Question

if a(4,3),b(-1,y) and c(3,4) are the vertices of a right triangle abc right angled at a then find the value of y.

Answer 1

Question:

If A(4,3), B(-1,y) and C(3,4) are the vertices of a right triangle ABC, right angled at A, then find the value of y.

Answer:

First we have to find the lengths of sides of the triangle ABC, i.e., distance between each two points, AB, BC and AC.

AB = √((4 - (- 1))² + (3 - y)²)

AB = √(5² + (3 - y)²)

AB = √(25 + 9 - 6y + y²)

AB = √(y² - 6y + 34)

AB² = y² - 6y + 34

BC = √((- 1 - 3)² + (y - 4)²)

BC = √((- 4)² + (y - 4)²)

BC = √(16 + y² - 8y + 16)

BC = √(y² - 8y + 32)

BC² = y² - 8y + 32

AC = √((4 - 3)² + (3 - 4)²)

AC = √(1² + (-1)²)

AC = √(1 + 1)

AC = √2

AC² = 2

Given that ΔABC is right angled at A. Thus BC is the hypotenuse.

According to Pythagoras' Theorem, AC² can be written as,

AC² = BC² - AB²

2 = (y² - 8y + 32) - (y² - 6y + 34)

2 = y² - 8y + 32 - y² + 6y - 34

2 = - 2y - 2

- 1 × - 2  =  - 1 (2y + 2)

- 2 = 2y + 2

- 4 = 2y

- 2 = y

Thus the value of y is  - 2.