Geometry for CAT/XAT/SSCCGL/GATE/GRE/GMAT/IPM/NPAT

Explore the basics of triangles, including the three-sided definition and the interior angle sum of 180 degrees, and learn the exterior angle property and vertical opposite angles.

Learn the triangle inequality: the sum of any two sides exceeds the third, and the difference of two sides is less than the third, illustrated with right triangles.

Identify and classify triangles by angles into right, acute, and obtuse types, and by sides into equilateral, isosceles, and scalene, using Pythagoras theorem.

Examine triangle types and side relationships, proving isosceles cases, and apply triangle inequality and pythagoras to determine integer third sides for 6 and 8 cm triangles, including obtuse cases.

Apply the exterior angle theorem to triangles, including isosceles cases, to find exterior angle measures by using interior angles and the sum of remote interior angles.

Apply the Pythagoras theorem to solve right-triangle problems, identify the hypotenuse and legs, and compute ladder distances between two buildings as well as areas of squares.

Learn to derive pythagorean triplets from odd bases 3,5,7,9 and even bases using simple formulas, producing 3-4-5, 5-12-13, 7-24-25, and 9-40-41 with twenty-five practice triplets.

Apply Pythagorean triplets to geometry problems, using triples like 3-4-5, 5-12-13, and 7-24-25 and their multiples. Solve rectangle in circle and radius-based questions efficiently.

Explore similarity and congruence in triangles, recognizing that similar triangles share shape with proportional sides and equal corresponding angles, while congruent triangles have identical size and shape.

Explore basic practice questions to understand similarity and congruence in triangles. Learn how angle-angle-side and side-angle-side criteria establish correspondence and determine whether triangles are similar or congruent.

Identify how equal angles yield similar triangles and use corresponding side ratios to find missing lengths and areas.

Learn to apply similarity in triangles to solve area and length problems, using correspondence and parallel lines, and explore the basic proportionality theorem and midpoint concepts.

Apply the basic proportionality theorem to triangles, use parallel lines and similar triangles to derive side ratios and area relations from midpoint constructions and problem examples.

Explore the centroid and medians of a triangle, apply Apollonius theorem to relate side and median squares, and use the identity m_a^2+m_b^2+m_c^2=3/4(a^2+b^2+c^2) to solve for side sums.

Explore how circumcenter and incenter define key triangle centers by using angle bisectors and perpendicular bisectors to derive the inradius and basic circle properties inside the triangle.

Explore special triangles—30-60-90 and 45-45-90—along with equilateral triangles, and learn centroid, medians, altitudes, perpendicular bisectors, circumcenter, incenter, and area formulas.

this lecture covers triangle area formulas, including base times height over two, area via two sides and the included angle, abc/(4R) with circumradius, and the s(s-a)(s-b)(s-c) method, with 3-4-5 examples.

Learn how to calculate areas of triangles in various configurations, including right triangles, isosceles cases, and circumcircle scenarios, using base-height and radius relationships.

Derive formula for the area of an isosceles triangle by dropping the altitude to the base, yielding height h = sqrt(a^2 - b^2/4) and area = 1/2 base times height.

Observe that a triangle's area is proportional to base for fixed height, and to height for fixed base. Divide the triangle by its medians into six equal-area parts.

Apply area proportionality to triangle geometry by showing that medians partition a triangle into six equal areas, and derive area ratios using similar triangles and midpoints for related shapes.

Solve the sum of the interior angles for two regular polygons with A and B sides, where B equals twice A and B’s interior angle is 3/2 of A’s.

Explore parallelograms, their properties, and special cases like rectangles, rhombuses, and squares, including area as base times height and diagonals that bisect each other.

Master the geometry of quadrilaterals by examining congruent triangles, diagonals, and angle relations; learn area formulas for triangles and trapeziums using base, height, and parallel sides.

Explore how diagonals, equal angles, and midpoints reveal when a quadrilateral is a square, rectangle, or rhombus, using proportionality, parallel lines, and midpoint theorems.

Explore the basics of trigonometry in right triangles, including sine, cosine, and tangent ratios, reciprocal identities, the Pythagorean theorem, and the unit circle with standard values.

Apply the sine and cosine rules to triangles to compute area and relate sides to angles, and solve a sample problem with sides 8, 6, and 4 to find DC.

Explore heights and distances by applying angle of elevation and angle of depression to compute building and tower heights from an observer, using right triangle trigonometry.

Solve heights and distances problems using angles of elevation and depression, illustrated by two lampposts across a 50-meter road and related computation of distances and heights.

Apply trigonometry to a 100-meter mountain to relate the cannon’s sight lines with soldier distances using angles of depression. Compute X1, X2, and related distances via tangent and angle complements.

Explore tangents to circles, including tangents from external points and the radius perpendicular to tangent, with formulas for external and transverse tangents: sqrt(d^2-(r1-r2)^2) and sqrt(d^2-(r1+r2)^2).

Explore tangents to circles from exterior and interior points, determine tangent lengths, and analyze common tangents between two circles using center distances and similar triangles.

Explains the theorem on chords of the circle, showing inscribed angles subtend the same arc and are equal, using similar triangles and tangent ideas.

Explore the theorem on chords of the circle through problem solving with exterior angles, isosceles triangles, and tangent properties; apply cosine rule and similarity to circle and square configurations.

Explore cyclic quadrilaterals: prove opposite angles are supplementary, use exterior angle equals remote interior, and apply area formulas for inscribed quadrilaterals.

Learn cyclic quadrilaterals and inscribed angles on circles, with example angles 96 and 48 degrees, and explore relations among opposite sides, diagonals, and acute triangle angles.

Explore mensuration and 3d geometry with cuboids and cubes, covering volume, total surface area, lateral surface area, body diagonals, and diagonal applications in packing and painting.

Compute the volume and surface area of 13 cubes with edge 2 cm, yielding 104 cm³ and 176 cm², then compare to a six-cube arrangement using cube-root ratios.

Learn prisms with regular polygon bases and cylinders; apply volume as base area times height and surface area as twice base area plus base perimeter times height, with hole-through examples.

Calculate the volume of a cylindrical drum from its radius and height and analyze the surface area of hollow and open cylinders within the 3d geometry of prisms and cylinders.

Explore three-dimensional geometry of pyramids and cones with regular polygon bases, deriving height, base area, and volume formulas, and applying surface area, cone ratios, and practical problems.

Explore mensuration and 3d geometry by solving pyramid and cone problems, including square-based pyramids, altitudes, hypotenuse, and volume ratios of cone slices.

Explore the sphere in three-dimensional space as the locus of points equidistant from a point, derive its surface area and volume, compare to cylinders, and relate painting cost to area.

Explore mensuration and 3d geometry with hemispheres and cylinders, solving volume and surface area, and analyzing a ball atop a hollow cylinder to find the top distance from the base.

solve the missing area puzzle by forming similar triangles, using proportional areas and algebra to transform a geometric figure into linear equations that yield the target area.

Apply the midpoint theorem and basic proportionality in triangles, using parallel lines to create proportional segments on sides and identify midpoint relationships.

This lecture shows how perpendicular medians partition a triangle into six equal-area parts, uses median relationships to compute the total area, yielding 72 cm².

Explore two methods to compute the sum of the interior angles of a five-pointed star, using exterior angles and a triangular decomposition, revealing a total interior-angle sum of 540 degrees.

Using a right triangle with legs 300 ft and 400 ft, the pool's diameter is 500 ft, so the area is 62500 pi square feet.

Solve a trapezium-based angle problem by using similar triangles and transversal angles to show the angle at Q is a right angle, 90 degrees.

Explore the midpoint theorem and basic proportionality theorem in triangles by drawing lines parallel to a side, proving equal ratios on the other sides and solving length problems.