JIPMAT 2022 (QA) - Given below are two statements: Statement I: In A B C, A B=6 sqrt(3) ~cm, A C=12 ~cm and B C=6 ~cm, then angle B=90^ Statement II: In A B C, is an isosceles with A C=B C. If A B^2=2 AC^2, Then angle C=90^ In the light of the above statement, choose the correct answer form the question below. | PYQs + Solutions

JIPMAT 2022

Geometry

Triangles

Easy

Given below are two statements:
Statement I: In $\triangle A B C, A B=6 \sqrt{3} \mathrm{~cm}, A C=12 \mathrm{~cm}$ and $B C=6 \mathrm{~cm}$ , then angle $B=90^{\circ}$
Statement II: In $\triangle A B C$ , is an isosceles with $A C=B C$ . If $A B^{2}=2 AC^{2}$ , Then angle $C=90^{\circ}$
In the light of the above statement, choose the correct answer form the question below.

Both Statement I and Statement II are true

Both Statement I and Statement II are false

Statement I is true but Statement II is false

Statement I is false but Statement II is true

Correct Option: 1

Statement I:

\newline

Given:

AB = 6\sqrt{3}

cm,

AC = 12

cm,

BC = 6

Using Pythagorean theorem if angle B = 90°:

\newline

AB^2 + BC^2 = AC^2

(6\sqrt{3})^2 + 6^2 = 12^2

108 + 36 = 144

144 = 144

✓

Therefore Statement I is true.

Statement II:

\newline

Given isosceles triangle where

AC = BC

\newline

And

AB^2 = 2AC^2

In isosceles triangle, base angles are equal.

\newline

Let base angles each be

x

\newline

Then

C = 180° - 2x

AB^2 = 2AC^2

, this means

\cos C = 0

\newline

Therefore

C = 90°

Therefore Statement II is also true.