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Geometry

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CAT 2025 Lesson : Lines & Triangles - Types of Triangles

3.2 Types of Triangles – Interior Angles

In a triangle, angles opposite to longer sides are larger in value. Accordingly, triangles are classified as acute, obtuse or right-angled using the following properties of their longest side or largest angle.

Definition	Figure
Acute Angled Triangle Angle: All angles of the triangle are acute, i.e. less than $90\degree$ . Sides: Where $a$ is the longest side, $a^{2} < b^{2} + c^{2}$
Obtuse Angled Triangle Angle: Two of the angles are acute, i.e. less than $90\degree$ . The third angle is obtuse, i.e. between $90\degree$ and $180\degree$ . Sides:Where $a$ is the longest side, $a^{2} > b^{2} + c^{2}$
Right Angled Triangle Angle: Two of the angles are acute, i.e. less than $90\degree$ . The third angle is right-angled, i.e. equals $90\degree$ . Sides:Where $a$ is the longest side, $a^{2} = b^{2} + c^{2}$

Also the following altitude properties apply for Acute angles and Obtuse angles of triangles. Note that in the case of right-angled triangles, two of the sides serve as each other's altitudes and is hence not necessary.

Definition	Figure
When the altitude AD lies inside the triangle, then AC $^{2}$ $=$ AB $^{2}$ $+$ BC $^{2}$ $-$ $( 2 \times$ BD $\times$ BC)
When the altitude AD lies outside the triangle, then AC $^{2}$ $=$ AB $^{2}$ $+$ BC $^{2}$ $+ ( 2 \times$ BD $\times$ BC)

Example 9

What is the number of acute-angled triangles that have sides of length

4

cm,

6

cm and

x

cm, where

x

is an integer?

Solution

Sum of any

2

sides is greater than the third side.

x + 4 > 6

⇒

x > 2

6 + 4 > x

⇒

x < 10

Possible integral values of

x

are

3, 4, 5, 6, 7, 8

and

9

.

However the triangle is acute when

a^{2} < b^{2} + c^{2}

, where

a

is the longest side.

This happens only when x can take the following

\bm{3}

values.

x = 5

(6^{2} < 4^{2} + 5^{2})

6

(6^{2} < 4^{2} + 6^{2})

7

(7^{2} < 4^{2} + 6^{2})

Answer:

3

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