Question

A(6, –3), В(0, 5) and C(–2, 1) are vertices of ΔАВС. Points P(3, 1) and Q(2, –1) lie on sides AB and AC respectively. Check whether `(AP)/(PB) = (AQ)/(QC)`.

Answer 1

Given: The vertices of ΔABC are A(6, –3), В(0, 5) and C(–2, 1). 

Points P(3, 1) and Q(2, –1) lies on sides AВ and AC respectively.

To check: `(AP)/(PB) = (AQ)/(QC)`

By using distance formula,

AP = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

= `sqrt((3 - 6)^2 + [1 - (-3)]^2`

= `sqrt((-3)^2 + (1 + 3)^2`

= `sqrt(9 + (4)^2`

= `sqrt(9 + 16)`

= `sqrt(25)`

= 5   ...(1)

PB = `sqrt((0 - 3)^2 + (5 - 1)^2`

= `sqrt((-3)^2 + (4)^2`

= `sqrt(9 + 16)`

= `sqrt(25)`

= 5   ...(2)

AQ = `sqrt((2 - 6)^2 + [-1 - (-3)]^2`

= `sqrt((-4)^2 + (-1 + 3)^2`

= `sqrt(16 + (2)^2`

= `sqrt(4 + 16)`

= `sqrt(20)`

= `2sqrt(5)`   ...(3)

QC = `sqrt([(-2) - 2]^2 + [1 - (-1)]^2`

= `sqrt((-4)^2 + (2)^2`

= `sqrt(16 + 4)`

= `sqrt(20)`

= `2sqrt(5)`   ...(4)

Putting the value of AP, PB, AQ and QC is

`(AP)/(PB) = (AQ)/(QC)`

⇒ `5/5 = (2sqrt(5))/(2sqrt(5))`

⇒ 1 = 1

So, in the triangle ABС:

`(AP)/(PB) = (AQ)/(QC)`

Hence Proved.