Unlocking Geometry: A Comprehensive Guide for All

A Comprehensive Guide for Grade 10 and Beyond, including Beginners, Middle School, and High School Students.

Created byLakshmi Narayana

Last updated 6/2023

English

What you'll learn

Triangles- Similarities, Thales Theorem and it applications, Applications of Converse of Basic Proportionality Theorem & its applications.
Triangles- Criteria for Similarity of Triangles & related proofs, Areas of Similar Triangles & its applications.
Triangles- Pythagoras Theorem and its Applications, and Converse of Pythagoras Theorem and its Applications.
Circles - Tangent to a Circle, Number of Tangents from a Point on a Circle and Their Length, Properties of Tangents, Secant and Intersecting Lines.
Constructions - Division of a line Segment, Construction of a Triangle Similar to a Given Triangle, Construction of Tangents to a Circle.
At the end of the course, students will not only have learned about the topics in detail but also be able to solve various problems based on them.

Course content

3 sections • 183 lectures • 22h 3m total length

Session 1 - Introduction9:34
Introduction & Similar Figures
Session 1 - Q 15:14
In Figure, if ∆ABC ~ ∆DEF and their sides are of lengths (in cm)
as marked along them, then find the lengths of the sides of each triangle.
Session 1 - Q 24:25
In the given figure, △ABC ~△PQR. Find the value of y + z.
Session 1 - Q 34:04
If in two triangles DEF and PQR, ∠D = ∠Q and ∠R = ∠E,
then which of the following is not true?
EF/PR = DF/PQ
DE/PQ = EF/RP
DE/QR = DF/PQ
EF/RP = DE/QR
Session 1 - Q 46:39
It is given that ABC ~ DFE, ∠A = 30°, ∠C = 50°, AB = 5cm,
AC = 8 cm and DF= 7.5 cm. Then, the following is true:
DE = 12 cm, ∠F = 50°
DE = 12 cm, ∠F = 100°
EF = 12 cm, ∠D = 100°
EF = 12 cm, ∠D = 30°
Test your understanding
Home Assignment
Session 2 - Introduction12:36
Basic Proportionality Theorem (BPT)& Its applications
Session 2 - Q 14:57
If a line intersects sides AB and AC of a ∆ABC at D and E respectively and is parallel to BC, prove that (AD )/AB = (AE )/AC.
Session 2 - Q 26:32
ABCD is a trapezium with AB || DC. E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB. Show that AE/ED = BF/FC
Session 2 - Q 34:37
In Fig. DE∥BC and CD∥EF. Prove that AD2 = AB × AF.
Session 2 - Q 47:29
In △ABC, D and E are points on the sides AB and AC respectively, such that DE || BC. If AD = 4x - 3, AE = 8x - 7, BD = 3x - 1 and CE = 5x -3 , find the value of x.
Test your understanding
Home Assignment
Session 3 - Introduction7:36
Converse of Basic Proportionality Theorem & its applications
Session 3 - Q 15:38
In figure, PS/SQ = PT/TR and ∠PST = ∠PRQ.
Prove that PQR is an isosceles triangle.
Session 3 - Q 24:30
If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium.
Session 3 - Q 35:52
In the given figure, ∠A = ∠B and AD = BE. Show that DE ∥ AB.
Session 3 - Q 43:09
If D and E are points on the respective sides AB and AC. △ABC such that, AD = 6 cm, BD = 9 cm, AE = 8 cm, EC = 12 cm. Prove that DE || BC.
Test your understanding
Home Assignment
Session 4 - Introduction1:59
Criteria for Similarity of Triangles & related proofs.
• AAA Similarity Criterion
• SSS Similarity Criterion
• SAS Similarity Criterion
Problems Based on Similarity of Triangles
Session 4 - Theorem 110:19
1. AAA Similarity Criterion:

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio(or proportion) and hence the two triangles are similar.
Session 4 - Theorem 29:31
2. SSS Similarity Criterion:
If in two triangles, sides of one triangle are proportional to (i.e. in the same ratio ) to the side of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.
Session 4 - Theorem 36:39
3. SAS Similarity Criterion
If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then two triangles are similar.
Session 4 - Q 12:53
In the adjoining figure, △AHK is similar to △ABC.
If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find AC.
Session 4 - Q 24:29
Observe the figure and then find ∠P.
Session 4 - Q 37:22
A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.
Test your understanding
Home Assigment
Session 5 - Introduction2:40
Proofs Based on Similarity of Triangles:
Session 5 - Q 13:35
In figure, if PQ || RS, prove that ∆POQ ~ ∆SOR.
Session 5 - Q 23:50
In figure, OA. OB = OC. OD. Show that ∠A = ∠C and ∠B = ∠D.
Session 5 - Q 313:42
In figure, CM and RN are respectively the medians of ∆ABC and ∆PQR. If ∆ABC ~ ∆PQR, prove that:
∆AMC ~ ∆PNR
(CM )/RN = (AB )/PQ
∆CMB ~ ∆RNQ
Session 5 - Q 46:45
In the given figure, AB ∥ PQ ∥ CD, AB = x, CD = y, PQ = z
Prove that (1 )/(x ) + (1 )/(y ) = (1 )/(z )
Test your understanding
Home Assignment
Session 6 - Introduction9:53
Areas of Similar Triangles &
Problems Based on Areas of Similar Triangles
Session 6 - Q 18:31
In figure, the line segment XY is parallel to side AC of ∆ABC and it divides the triangle into two parts of equal areas. Find the ratio AX/AB
Session 6 - Q 26:51
In the given figure, PA/AQ = PB/BR = 3. If the area △PQR is 32 cm2,
then find the area of the quadrilateral AQRB.
Session 6 - Q 32:56
ΔABC and ΔDEF are similar and AB = 1/3 DE,
then find ar(ΔABC) : ar(ΔDEF)
Session 6 - Q 410:42
In the given figure, if DE∥BC and AD : DB = 5 : 4, then find (ar(△DFE) )/(ar(△CFB) )
Test your understanding
Home Assignment
Session 7 - Introduction1:22
Areas of Similar Triangles
&
Proofs Based on Areas of Similar Triangles
Session 7 - Q 14:17
If △ABC ∼ △PQR and AD and PS are bisectors of corresponding angles A and P, then prove that (ar(ΔABC) )/(ar(ΔPQR)) = AD^2/PS^2 .
Session 7 - Q 23:37
If the area of two similar triangles are equal, prove that they are congruent.
Session 7 - Q 33:55
Diagonals of a trapezium PQRS intersect each other at the
point O, PQ ∥ RS and PQ = 3RS. Find the ratio of the areas
of triangles △POQ and △ROS.
Session 7 - Q 44:16
Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.
Test your understanding
Home Assignment
Session 8 - Introduction6:35
Pythagoras Theorem and its Applications
Session 8 - Q 15:42
In figure, ∠ACB = 900 and CD⊥ AB. Prove that (BC^2)/(AC^2 ) = BD/AD
Session 8 - Q 23:56
In figure, if AD ⊥ BC, prove that AB2 + CD2 = BD2 + AC2.
Session 8 - Q 35:24
BL and CM are medians of a triangle ABC right angled at A.
Prove that 4(BL2 + CM2) = 5BC2.
Session 8 - Q 43:12
A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.
Test your understanding
Home Assignment
Session 9 - Introduction7:56
Proofs Based on Converse of Pythagoras Theorem
Session 9 - Q 15:55
O is any point inside a rectangle ABCD.
Prove that OB2 + OD2 = OA2 + OC2.
Session 9 - Q 28:21
ΔABC is right angled at C. If p is the length of the perpendicular from C to AB and a, b, c are the lengths of the sides opposite ∠A, ∠B and ∠C respectively, then prove that 1/P^2 = 1/a^2 + 1/b^2
Session 9 - Q 36:44
In a ΔABC, AD ⊥ BC and AD2 = BD × CD.
Prove that ΔABC is a right triangle.
Session 9 - Q 44:13
In an equilateral triangle of side √3 cm,
find the length of the altitude.
Test your understanding
Home Assignment
Exercise 1 - Q 14:16
Fill in the blanks using the correct word given in brackets :
(i) All circles are_______. (congruent, similar)
(ii) All squares are __________. (similar, congruent)
(iii) All _______ triangles are similar. (isosceles, equilateral)
(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are _________ and (b) their corresponding sides are _________. (equal, proportional)
Exercise 1 - Q 22:44
Give two different examples of pair of
(i) Similar figures. (ii) Non-similar figures.
Exercise 1 - Q 32:01
State whether the following quadrilaterals are similar or not
Exercise 2 - Q 16:12
In figure, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
Exercise 2 - Q 28:21
E and F are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether EF || QR :
(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
Exercise 2 - Q 35:02
In figure. 6.18, if LM || CB and LN || CD, prove that
(AM )/AB = (AN )/AD
Exercise 2 - Q 46:07
In figure, DE || AC and DF || AE. Prove that (BF )/FE = (BE )/EC
Exercise 2 - Q 57:49
In figure, DE || OQ and DF || OR. Show that EF || QR.
Exercise 2 - Q 66:24
In figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
Exercise 2 - Q 75:37
Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).
Exercise 2 - Q 85:59
Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in
Class IX).
Exercise 2 - Q 99:54
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that (AO )/BO = (CO )/DO
Exercise 2 - Q 1012:09
The diagonals of a quadrilateral ABCD intersect each other at the point O such that (AO )/BO = (CO )/DO Show that ABCD is a trapezium.
Exercise 3 - Q 115:09
State which pairs of triangles in figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :
Exercise 3 - Q 28:31
In figure, ∆ODC ~ ∆OBA, ∠BOC = 1250 and ∠CDO = 700. Find ∠DOC, ∠DCO and ∠OAB.
Exercise 3 - Q 37:23
Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that
(OA )/OC = (OB )/OD
Exercise 3 - Q 48:27
In figure, (QR )/QS = (QT )/PR and ∠1 = ∠2. Show that ∆PQS ~ ∆TQR.
Exercise 3 - Q 53:56
S and T are points on sides PR and QR of ∆PQR such that ∠P = ∠RTS. Show that ∆RPQ ~ ∆RTS.
Exercise 3 - Q 66:44
In figure, if ∆ABE = ∆ACD, show that ∆ADE ~ ∆ABC.
Exercise 3 - Q 714:15
In figure, altitudes AD and CE of ∆ABC intersect each other at the point P. Show that:
(i) ∆AEP ~ ∆CDP
(ii) ∆ABD ~ ∆CBE
(iii) ∆AEP ~ ∆ADB
(iv) ∆PDC ~ ∆BEC
Exercise 3 - Q 85:40
E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ABE ~ ∆CFB.
Exercise 3 - Q 96:41
In figure, ABC and AMP are two right triangles, right angled at B and M
respectively. Prove that:
(i) ∆ABC ~ ∆AMP
(ii) (CA )/PA = (BC )/MP
Exercise 3 - Q 1014:37
CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ~ ∆FEG, show that:
(i) (CD )/GH = (AC )/FG
(ii) ∆DCB ~ ∆HGE
(iii) ∆DCA ~ ∆HGF
Exercise 3 - Q 115:12
In figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD⊥BC and EF⊥AC, prove that ∆ABD ~ ∆ECF.
Exercise 3 - Q 129:34
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ∆PQR (see figure). Show that ∆ABC ~ ∆PQR.
Exercise 3 - Q 136:07
D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show
that CA2 = CB. CD.
Exercise 3 - Q 1419:11
Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.
Show that ∆ABC ~ ∆PQR.
Exercise 3 - Q 156:05
A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
Exercise 3 - Q 169:46
If AD and PM are medians of triangles ABC and PQR, respectively where
∆ABC ~ ∆PQR, prove that (AB )/PQ = (AD )/PM
Exercise 4 - Q 15:36
Let ∆ABC~∆DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
Exercise 4 - Q 27:03
Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.
Exercise 4 - Q 36:38
In figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that (ar (ABC) )/(ar (DBC)) = (AO )/DO
Exercise 4 - Q 45:30
If the areas of two similar triangles are equal, prove that they are congruent.
Exercise 4 - Q 57:48
D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ABC. Find the ratio of the areas of ∆DEF and ∆ABC.
Exercise 4 - Q 68:11
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Exercise 4 - Q 79:09
Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
Exercise 4 - Q 83:39
ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is
(A) 2 : 1 (B) 1 : 2 (C) 4 : 1 (D) 1 : 4
Exercise 4 - Q 91:58
Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio
(A) 2 : 3 (B) 4 : 9 (C) 81 : 16 (D) 16 : 81
Exercise 5 - Q 19:59
Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
(i) 7 cm, 24 cm, 25 cm
(ii) 3 cm, 8 cm, 6 cm
(iii) 50 cm, 80 cm, 100 cm
(iv) 13 cm, 12 cm, 5 cm
Exercise 5 - Q 26:39
PQR is a triangle right angled at P and M is a point on QR such that PM⊥QR. Show that PM2 = QM. MR.
Exercise 5 - Q 39:47
In figure, ABD is a triangle right angled at A and AC⊥BD. Show that
(i) AB2 = BC. BD
(ii) AC2 = BC. DC
(iii) AD2 = BD. CD
Exercise 5 - Q 43:27
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
Exercise 5 - Q 55:52
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
Exercise 5 - Q 65:37
ABC is an equilateral triangle of side 2a. Find each of its altitudes.
Exercise 5 - Q 77:23
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
Exercise 5 - Q 813:31
In Fig. 6.54, O is a point in the interior of a triangle ABC, OD⊥BC, OE⊥AC and OF⊥AB. Show that
(i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2,
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
Exercise 5 - Q 94:25
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.
Exercise 5 - Q 106:03
A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
Exercise 5 - Q 1111:18
An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after 11/2 hour?
Exercise 5 - Q 125:17
Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
Exercise 5 - Q 137:13
D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2.
Exercise 5 - Q 147:59
The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3 CD (see Fig. 6.55). Prove that 2AB2 = 2AC2 + BC2.
Exercise 5 - Q 1513:28
In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3BC. Prove that 9AD2 = 7AB2.
Exercise 5 - Q 167:12
In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
Exercise 5 - Q 172:51
Tick the correct answer and justify: In ∆ABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm. The angle B is :
(A) 1200 (B) 600
(C) 900 (D) 450
Exercise 6 - Q 18:45
In figure, PS is the bisector of ∠QPR of ∆PQR. Prove that QS/SR = PQ/PR
Exercise 6 - Q 215:23
In figure, D is a point on hypotenuse AC of ∆ABC, such that BD⊥AC, DM⊥BC and DN⊥AB. Prove that :
(i) DM2 = DN. MC (ii) DN2 = DM. AN
Exercise 6 - Q 36:25
In figure, ABC is a triangle in which ∠ABC > 90° and AD⊥CB produced. Prove that
AC2 = AB2 + BC2 + 2BC. BD.
Exercise 6 - Q 45:20
In figure, ABC is a triangle in which ∠ABC < 90° and AD⊥BC. Prove that
AC2 = AB2 + BC2 – 2BC. BD.
Exercise 6 - Q 58:44
In figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:
(i) AC2 = AD2 + BC. DM + (BC/2)^2
(ii) AB2 = AD2 – BC. DM + (BC/2)^2
(iii) AC2 + AB2 = 2 AD2 + 1/2BC2
Exercise 6 - Q 610:08
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
Exercise 6 - Q 74:07
In figure, two chords AB and CD intersect each other at the point P. Prove that : (i) ∆APC ~ ∆DPB (ii) AP. PB = CP. DP
Exercise 6 - Q 85:45
In figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) ∆PAC ~ ∆PDB
(ii) PA. PB = PC. PD
Exercise 6 - Q 99:26
In figure, D is a point on side BC of ∆ABC such that BD/CD = AB/AC Prove that AD is the bisector of ∠BAC.
Exercise 6 - Q 108:55
Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (see Fig. 6.64)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?

Session 1 - Introduction11:23
Introduction
Session 1 - Theorem 15:28
The TANGENTat any point of a circle is perpendicular to the radius through the point of contact.
Session 1 - Q 13:05
From a point P, 10 cm away from the centre of a circle, a tangent PT of length 8 cm is drawn. Find the radius of the circle.
Session 1 - Q 23:37
In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Find the radius of the circle.
Test your understanding
Home Assignment
Session 2 - Introduction6:53
Number of Tangents from a Point on a Circle
Session 2 - Theorem 23:35
The lengths of two tangents drawn from an external point to a circle are equal
Session 2 - Q 13:00
Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
Session 2 - Q 26:04
In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm then find the radius of the smaller circle.
Session 2 - Q 33:59
Two concentric circles are of radii 7 cm and r cm respectively where r > 7. A chord of the larger circle of the length 48 cm, touches the smaller circle. Find the value of r.
Session 2 - Q 47:12
PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Figure). Find the length TP.
Test your understanding
Home Assignment
Session 3 - Introduction2:44
Number of Tangents from a Point on a Circle :
Problems based on length of the tangent from an external point to a circle
Session 3 - Q 18:06
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2 ∠OPQ.
Session 3 - Q 25:33
If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60o. Find the length of chord AB.
Session 3 - Q 33:33
If two tangents inclined at an angle of 600 are drawn to a circle of a radius 3 cm, then find the length of each tangent.
Session 3 - Q 44:38
In two concentric circles, prove that all chord of the outer circle which touch the inner circle are of equal length.
Test your understanding
Home Assignment
Session 4 - Introduction2:27
Miscellaneous problemsrelated to circles
Session 4 - Q 18:29
In fig. the radius of the in circle of △ABC of area 84 cm2 and the lengths of the segments AP and BP into which side AB is divided by the point of contact are 6 cm and 8 cm. Find the lengths of the sides AC and BC
Session 4 - Q 27:10
If a, b and c are the sides of a right angled triangle, where c is hypotenuse, then prove that the radius of the circle which touches the sides of the triangle is given by r = (a+b-c)/2
Session 4 - Q 35:30
A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.
Session 4 - Q 46:48
In fig., a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, P and S respectively. If AB = 29 cm, AD = 23 cm, ZB = 90o and DS = 5 cm, then find the radius of the circle (in cm.)
Test your understanding
Home Assignment
Exercise 1 - Q 11:20
How many tangents can a circle have?
Exercise 1 - Q 22:32
Fill in the blanks :
(i) A tangent to a circle intersects it in ___________ point (s).
(ii) A line intersecting a circle in two points is called a _____________.
(iii) A circle can have ___________ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called __________.
Exercise 1 - Q 33:19
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm.
Exercise 1 - Q 41:44
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Exercise 2 - Q 14:41
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm (B) 12 cm
(C) 15 cm (D) 24.5 cm
Exercise 2 - Q 23:17
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that
∠POQ = 110°, then ∠PTQ is equal to
(A) 60° (B) 70° (C) 80° (D) 90°
Exercise 2 - Q 39:10
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to
(A) 50° (B) 60° (C) 70° (D) 80°
Exercise 2 - Q 46:03
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Exercise 2 - Q 57:16
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Exercise 2 - Q 64:59
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
Exercise 2 - Q 77:03
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Exercise 2 - Q 87:17
A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that
AB + CD = AD + BC
Exercise 2 - Q 910:15
In Fig. 10.13, XY and X’Y’ are two parallel tangents to a circle with centre O and
another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°
Exercise 2 - Q 105:34
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
Exercise 2 - Q 116:57
Prove that the parallelogram circumscribing a circle is a rhombus.
Exercise 2 - Q 1217:00
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.
Exercise 2 - Q 139:40
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Session 1 - Introduction10:51
Introduction
Session 1 - Q 18:13
Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB such that AP/AB = 3/5
Session 1 - Q 27:39
Draw a line segment of length 5 cm and divide it in the ratio 3 : 7.
Session 1 - Q 38:21
Draw a line segment of length 8 cm and divide it internally in the ratio 4 : 5.
Test your understanding
Home Assignment
Session 2 - Introduction6:29
Construction of a Triangle Similar to Given Triangle when m/n < 1 or m < n
Session 2 - Q 110:59
Construct a triangle similar to a given triangle ABC with its sides equal to 3/4 of the corresponding sides of the triangle ABC (i.e., of scale factor 3/4).
Session 2 - Q 210:29
Construct an isosceles triangle ABC with base BC = 6 cm, AB = AC and ∠A = 900. Draw another similar triangle whose sides are 4/5 times of the sides of △ABC. Justify your construction.
Session 2 - Q 37:02
Construct a triangle with BC = 7cm, ∠B = 450 and ∠C = 600.Then, construct a similar triangle to its whose sides are 3/5 times of the corresponding sides of the given triangle.
Test your understanding
Home Assignment
Session 3 - Introduction6:10
Construction of a Triangle Similar to Given Triangle when m/n> 1 or m > n
Session 3 - Q 110:45
Construct a triangle similar to a given triangle ABC with its sides equal to 5/3 of the corresponding sides of the triangle ABC (i.e., of scale factor 5/3).
Session 3 - Q 28:12
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 5 cm and 4 cm. Then construct another triangle whose sides are 5/3 times the corresponding sides of the given triangle.
Session 3 - Q 39:29
Draw a triangle ABC with side BC = 7 cm, ∠B = 450, ∠A = 1050. Then construct a triangle whose sides are 4/3 times the corresponding sides of ∆ABC.
Test your understanding
Home Assignment
Session 4 - Introduction2:06
Construction of Tangents to a Circle
Session 4 - Q 14:30
Draw a circle of diameter AB = 6 cm with centre O and then draw a tangent to the circle at point A or B.
Session 4 - Q 26:02
Draw a circle of radius 4 cm from a point P, 7 cm from the centre of the circle, draw a pair of tangents to the circle measure the length of each tangent segment.
Session 4 - Q 35:25
Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on the outer circle, construct the pair of tangents to the inner circle.
Session 4 - Q 47:11
Draw a circle of radius 4 cm. Take two points P and Q on one of its extended diameters, each at a distance of 9 cm from its centre. Draw tangents to the circle from these two points P and Q.
Session 4 - Q 57:27
Draw a pair of tangents to a circle of radius 3 cm which are inclined to each other at angle of 60°.
Test your understanding
Home Assignment
Session 5 - Introduction2:36
Construction of a Quadrilateral Similar to a Given Quadrilateral
Session 5 - Q 110:04
Construct a quadrilateral AB = 3 cm, BC = 4 cm , AC = 5 cm, CD = 2 cm and ∠A = 700 . And construct a quadrilateral similar to the given quadrilateral with scale factor 3/5.
Test your understanding
Home Assignment
Exercise 1 - Q 112:29
Draw a line segment of length 7.6 cm and divide it in the ratio 5: 8. Measure the two parts.
Exercise 1 - Q 214:13
Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are 2/3 of the corresponding sides of the first triangle.
Exercise 1 - Q 316:05
Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are 7/5 of the corresponding sides of the first triangle.
Exercise 1 - Q 413:57
Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 11/2 times the corresponding sides of the isosceles triangle.
Exercise 1 - Q 516:09
Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 3/4 of the corresponding sides of the triangle ABC.
Exercise 1 - Q 615:58
Draw a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then, construct a
triangle whose sides are 4/3 times the corresponding sides of D ABC.
Exercise 1 - Q 715:08
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are 5/3 times the corresponding sides of the given triangle.
Exercise 2 - Q 110:31
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.
Exercise 2 - Q 211:58
Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.
Exercise 2 - Q 311:30
Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.
Exercise 2 - Q 411:13
Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.
Exercise 2 - Q 58:47
Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.
Exercise 2 - Q 610:52
Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠B = 90°. BD is the
perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to this circle.
Exercise 2 - Q 79:31
Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.

Requirements

Basic knowledge of fundamental geometry and using of basic usage of mathematical instruments – like ruler, protractor, compass, etc.
Each chapter includes prerequisite knowledge classes in which the child gains extensive knowledge and a thorough understanding of the chapters.
Basic elementary math knowledge.

Description

This course is designed for all middle school and high school students. This course is intended for students under 18 may use the services only if a parent or guardian opens their account, handles any enrolments, and manages their account usage.
This course is carefully designed to explain various areas of Geometry.
It has 183 lectures spanning around 22 hours of on-demand videos that are divided into 3 sections, and each chapter is a section and further divided into simple sessions. The course is divided into a simplified day-by-day learning schedule.
Each topic is divided into simple sessions and explained extensively by solving multiple questions. Each session contains a detailed explanation of the concept.
An online test related to the concept for immediate assessment of understanding.
Session-based daily home assignments with a separate key. The students are encouraged to solve practise questions and quizzes provided at the end of each session.
This course will give you a firm understanding of the fundamentals and is designed in a way that a person with little or no previous knowledge can also understand it very well.
It covers 100% video solutions of various problems and situations.
Our design meets the real classroom experience by following classroom teaching practices. We have designed this course by keeping in mind all the needs of students and their desire to become masters in Geometry. This course is designed to benefit all levels of learners and will be the best gift for board-appearing students. Students love these easy methods and explanations. They enjoy learning math and never feel that math is troublesome.

Topics covered in the course:

Triangles
Circles
Constructions.

With this course you'll also get:

Perfect your mathematical skills on Geometry for better scores.

A Udemy Certificate of Completion is available for download.

Feel free to contact me with any questions or clarifications you might have.

I can't wait for you to get started on mastering the Geometry.

I look forward to seeing you on the course! :)

Benefits of Taking this Course:

On completion of this course, one will have detailed knowledge of Geometry and be able to easily solve all the problems, which can lead to scoring well in exams with the help of explanatory videos ensure complete concept understanding.

Downloadable resources help in applying your knowledge to solve various problems.

Quizzes help in testing your knowledge. In short, one can excel in math by taking this course.

Who this course is for:

This course has been designed for students of Grade 10th CBSE, ICSE, SSC, GCSE, IGCSE, SAT, ACT, GRE, and other board-appearing students.
Students studying for the public or other competitive examinations as well as job aspirants.
Home-school parents are looking for extra support with the fundamentals.
Anyone interested in revising or learning the basics of mathematics should.
Students in junior high and high school/secondary schools.
Anyone who wants to proficient mathematics and the solving different real life situations as well.
Anyone who wants to study math for fun after taking a break from school.
It will also benefit schools who wish to run classes in the absence of a teacher and make learning fun for their students.
It will also benefit teachers and schools who wish to improve their teaching skills and make learning fun for their students.
For 11th, 9th, and 8th grade students, this will help as a bridge course.
These are the people whose jobs require them to solve basic daily math-related problems.

Unlocking Geometry: A Comprehensive Guide for All

What you'll learn

Explore related topics

Course content

Triangles112 lectures • 12hr 57min

Circles37 lectures • 3hr 37min

Constructions34 lectures • 5hr 28min

Requirements

Description

Who this course is for: