Triangles

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A triangle is a three-sided polygon with three angles that always sum to $180°$. The side opposite the largest angle is the longest, and the side opposite the smallest angle is the shortest. Triangles are classified as acute (all angles < 90°), right-angled (one angle = 90°), or obtuse (one angle > 90°). By side length, they are equilateral (all sides equal, all angles 60°), isosceles (two sides equal, two angles equal), or scalene (all sides different).

Area formulas:

• General: $A = \frac{1}{2} \times \text{base} \times \text{height}$
• Heron’s formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$ (semi-perimeter)
• For equilateral triangle side $a$: $A = \frac{\sqrt{3}}{4}a^2$

Key properties:

• Triangle Inequality: Sum of any two sides must exceed the third side. For sides $a, b, c$: $a + b > c$, $b + c > a$, $c + a > b$
• Angle Bisector Theorem: The angle bisector divides the opposite side in ratio $AB:BC = \text{side}_1:\text{side}_2$
• Pythagoras Theorem (right-angled): $a^2 + b^2 = c^2$ where $c$ is the hypotenuse

⚡ CAT Tip: In CAT questions, look for hidden triangles in complex figures. Often the answer involves recognizing that a diagonal splits a quadrilateral into two triangles. For Pythagorean triplets, memorize: (3,4,5), (5,12,13), (7,24,25), (8,15,17), (9,40,41), (11,60,61) — these appear frequently and avoid square root calculations.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Congruence and Similarity

Two triangles are congruent if all three sides and all three angles are equal (SSS, SAS, ASA, AAS, RHS rules). Two triangles are similar if their corresponding angles are equal and sides are in proportion (AA, SSS, SAS criteria).

For similar triangles with sides in ratio $k$, their areas are in ratio $k^2$. This is crucial for solving many CAT problems involving shadows, poles, and mirror problems.

Special Triangle Results

1. Median: Joins a vertex to the midpoint of opposite side. All three medians intersect at centroid (dividing each median in 2:1 ratio from vertex).
2. Altitude: Perpendicular from vertex to opposite side. In right-angled triangle, altitude to hypotenuse $h = \frac{ab}{c}$ where $a, b$ are legs and $c$ is hypotenuse.
3. Angle Bisector: Divides opposite side proportionally to adjacent sides.
4. Circumradius $R = \frac{abc}{4A}$ and Inradius $r = \frac{A}{s}$

Right-Angled Triangle Tricks

For a right triangle with sides $a, b$ and hypotenuse $c$:

• $c = \sqrt{a^2 + b^2}$
• Area = $\frac{1}{2}ab$
• Inradius $r = \frac{a+b-c}{2}$
• Circumradius $R = \frac{c}{2}$

⚡ Common Mistake: Students often forget that the altitude to the hypotenuse creates two smaller triangles that are similar to each other AND to the original triangle. This gives three proportional relationships: $\frac{AD}{AB} = \frac{AB}{AC}$, $\frac{BD}{BC} = \frac{BC}{AC}$, and $\frac{AD}{BD} = \frac{AB^2}{BC^2}$.

Type of Questions Asked in CAT:

• Find area/perimeter given certain sides
• Prove triangle properties using congruence/similarity
• Problems involving medians, angle bisectors, altitudes
• Application in coordinate geometry (distance formula, area of triangle)

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

Advanced Theorems

1. Apollonius Theorem: $AB^2 + AC^2 = 2(AD^2 + BD^2)$ where $D$ is midpoint of $BC$

2. Stewart’s Theorem: For cevian $AD$ dividing $BC$ into $m$ and $n$: $AB^2 \cdot n + AC^2 \cdot m = BC(AD^2 + mn)$

3. Interior and Exterior Angle Bisectors: The exterior angle bisector divides the opposite side externally in ratio of adjacent sides.

4. Circumcircle and Incircle: In any triangle, $R \geq 2r$ with equality only for equilateral triangles. For equilateral triangle: $R = \frac{a}{\sqrt{3}}$ and $r = \frac{a}{2\sqrt{3}}$.

Coordinate Geometry Applications

Using coordinates:

• Area of triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$: $A = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$
• Centroid = $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$
• For incenter, use weighted average by sides

Trigonometric Approach

Using sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

Cosine rule: $c^2 = a^2 + b^2 - 2ab\cos C$

This is particularly useful when you know two sides and the included angle, or all three sides and want to find angles.

Previous Year CAT Pattern

Triangles appear in:

• Geometry section (3-4 questions per slot)
• Often combined with circles or quadrilaterals
• DILR sometimes includes triangle-based data

Problem-Solving Strategy

Step 1: Identify what type of triangle(s) you’re dealing with Step 2: List known quantities (sides, angles, area, perimeter) Step 3: Choose appropriate formula or theorem Step 4: Check if similarity/congruence can simplify the problem Step 5: Look for special triangles (right-angled, isosceles, equilateral)

⚡ Advanced Tip: For complex figures, try to find triangles with known ratios. Often extending lines or drawing altitudes creates simpler sub-problems. The centroid is particularly useful — it divides the triangle into 6 equal-area smaller triangles.

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📊 CAT Exam Essentials

Detail	Value
Sections	VARC (24 Qs), DILR (20 Qs), QA (22 Qs)
Time	2 hours (40 min per section)
Total	66 questions, 198 marks
Marking	+3 correct, −1 wrong (MCQ); no penalty for TITA
Mode	Computer-based, multiple sessions
Percentile	Normalized — 99+ needed for top IIMs

🎯 High-Yield Topics for CAT

• Reading Comprehension — 16-20 marks in VARC
• Para Summary + Odd Sentence — 8-12 marks
• DI Sets (Tables + Caselets) — 10-15 marks in DILR
• Arithmetic (Percentages + Profit/Loss) — 8-12 marks in QA
• Geometry + Mensuration — 6-10 marks
• Logarithm + Sequences — 6-10 marks

📝 Previous Year Question Patterns

• Q: “The passage is primarily concerned with…” [2024 VARC — RC passage]
• Q: “If f(x) = x² - 5x + 6, the value of f(3) is…” [2024 QA — Arithmetic]
• Q: “How many ways can 5 people be arranged around a round table…” [2024 DILR — Circular]

💡 Pro Tips

• VARC is the top priority — strong RC skills can push you to 99+ percentile quickly
• DILR: attempt 2 full sets out of 4-5 sets — accuracy matters more than coverage
• QA: arithmetic (time-speed-work) + geometry carry ~40% of QA marks
• Take 3-4 full mocks before the exam to find your section-wise pacing

🔗 Official Resources

• IIM CAT Official
• [CAT Syllabus](https://iimcat.ac.in/exam pattern)

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