34 days to CAT

Lesson 15 min· Section 1 of 4

The straight-up lesson, built for the exam.

Triangles

Formula sheet

Everything for this chapter, in one place. $a$ , $b$ , $c$ are the sides, $s = \dfrac{a+b+c}{2}$ .

Area, four ways


Base and height	$\dfrac{1}{2} \times \text{base} \times \text{height}$
Heron	$\sqrt{s(s-a)(s-b)(s-c)}$
Two sides and the angle between	$\dfrac{1}{2}ab\sin C$
Equilateral, side $a$	$\dfrac{\sqrt{3}}{4}a^2$

Right-angled triangles


Pythagoras	$a^2+b^2=c^2$
Altitude to the hypotenuse	$\dfrac{ab}{c}$
Largest that altitude can be	$\dfrac{c}{2}$ , when the triangle is isosceles
Inradius	$r = \dfrac{a+b-c}{2}$
Circumradius	$R = \dfrac{c}{2}$ — the hypotenuse is a diameter
Triples worth knowing	3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 20-21-29

Centres


Centroid	where the medians meet, splitting each 2 : 1 from the vertex
One median	cuts the triangle into two equal areas
All three medians	cut it into six equal areas
Two medians perpendicular, lengths $m_1, m_2$	area $= \dfrac{2}{3}m_1m_2$
Inradius	$r = \dfrac{\text{area}}{s}$
Circumradius	$R = \dfrac{abc}{4 \times \text{area}}$

Ratios of areas


Similar triangles	areas are in the square of the side ratio
$D$ on $AB$ and $E$ on $AC$	$\dfrac{[ADE]}{[ABC]} = \dfrac{AD}{AB} \times \dfrac{AE}{AC}$
Same height, different bases	areas are in the ratio of the bases
Midline	joins two midpoints, parallel to the third side and half its length
Medial triangle	area is $\dfrac{1}{4}$ of the original

Angles


Sine rule	$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R$
Cosine rule	$c^2 = a^2+b^2-2ab\cos C$
A side that is a diameter	the angle facing it is $90^\circ$
Exterior angle	equals the sum of the two opposite interior angles

Two results that answer whole questions


Viviani	in an equilateral triangle, the perpendiculars from any interior point to the three sides always add to the altitude
Triangle inequality	each side is less than the sum of the other two

Why this chapter matters

1

questions in a typical Quant section

10th

of 25 chapters

≈ 3 of 66 marks a section ride on it.

Asked in17 of 24 papers

20172025

Year by yearper section

2017: 12017: 1 per section
2018: 0.32018: 0.3 per section
2019: 0.72019: 0.7 per section
2020: 0.32020: 0.3 per section
2021: 12021: 1 per section
2022: 1.32022: 1.3 per section
2023: 12023: 1 per section
2024: 0.72024: 0.7 per section
2025: 12025: 1 per section

Question type22 asked

MCQ 54%Type-in 46%

Type-in answers carry no negative marking.

From every Triangles question in CAT papers since 2017.