IIM CAT 2026 cheat sheet for Geometry! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Essential Geometry for CAT 2026

Geometry is a core high-weightage topic in the Quantitative Aptitude section of the CAT examination. Mastering this section requires a firm grasp of core geometric properties, formulas, and structural relationships across lines, angles, triangles, circles, and polygons.

1. Lines, Angles, and Triangles

For any triangle with side lengths \(a\), \(b\), and \(c\), the triangle inequality theorem states that \(a + b > c\) and \(|a - b| < c\). Let \(s\) be the semi-perimeter \(s = \frac{a + b + c}{2}\).

• Heron's Formula: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)
• Inradius (\(r\)) and Circumradius (\(R\)): \(r = \frac{\text{Area}}{s}\) and \(R = \frac{abc}{4 \times \text{Area}}\)
• Pythagorean Theorem & Special Triangles: For right-angled triangles, \(a^2 + b^2 = c^2\). In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, sides are in the ratio \(1 : \sqrt{3} : 2\). In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, sides are in the ratio \(1 : 1 : \sqrt{2}\).

2. Circle Theorems & Intersecting Chords

Circles contribute significantly to CAT quantitative questions. Key chord and secant relationships include:

• Intersecting Chords Theorem: When two chords \(AB\) and \(CD\) intersect at point \(P\) inside a circle, \(PA \times PB = PC \times PD\).
• Tangent-Secant Theorem: If a tangent from point \(P\) touches the circle at \(T\) and a secant intersects at \(A\) and \(B\), then \(PT^2 = PA \times PB\).
• Alternate Segment Theorem: The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.

3. Polygons and Mensuration

For any convex polygon with \(n\) sides:

\[\text{Sum of Interior Angles} = (n - 2) \times 180^{\circ}\]\[\text{Number of Diagonals} = \frac{n(n - 3)}{2}\]