Question

IN triangle ABC, AB=AC and AB: BC = 2:1,
AN perpendicular toBC then show that 4AN² =15BC²​

Answer 1

Given:

In a triangle ABC,

• AB = AC
• AB : BC = 2 :1
• AN is perpendicular to BC

Observation:

• Given that BC = 2 : 1 and we know that AB is equal to AC, therefore, the ratio of AB : AC : BC = 2 : 2 : 1.
• Given that AB = AC, since both lengths are equal, it is an isosceles triangle.
• AN is perpendicular to BC, therefore AN = NB

Define variable:

Let the length of BC = x

AB = 2x

BC = 2x

* See attached for the drawn visual representation of the question.

Proof:

Triangle ABN is a right angle triangle, therefore we can apply Pythagoras Theorem which states that a² + b² = c²

$AN^2 + BN^2 = AB^2$

$AN^2 = AB^2 - BN^2$

$AN^2 = (2x)^2 - (\dfrac{1}{2}x)^2$

$AN^2 = 4x^2 - \dfrac{x^2}{4}$

$AN^2 = \dfrac{16x^2 - x^2}{4}$

$AN^2 = \dfrac{15x^2}{4}$

$4AN^2 = 15x^2$

$\text{We know that BC = x}$

$4AN^2 = 15BC^2 \text { (proved)}$