Question

Find the value of x such that the area of a triangle whose vertices have coordinates (6, 5), (8, 2) and (x, 11) is 15 square units.

Answer 1

`x =12\ or\ -8`

Explanation:

Given

`(x_1,y_1) = (6,5)`

`(x_2,y_2) = (8,2)`

`(x_3,y_3) = (x,11)`

`Area = 15`

Required

Find x

The area is calculated as thus:

`Area = (1)/(2)|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|`

Substitute values

`15 = (1)/(2)|6*(2 - 11) + 8*(11 - 5) + x(5 - 2)|`

`15 = (1)/(2)|6*(-9) + 8*(6) + x(3)|`

`15 = (1)/(2)|-54 + 48 + 3x|`

`15 = (1)/(2)|-6 + 3x|`

Multiply through by 2

`2 * 15 = (1)/(2)|-6 + 3x| * 2`

`30 = |-6 + 3x|`

Remove absolute sign

`30 = -6 + 3x\ or\ -30 = -6 + 3x`

Add 6 to both sides

`36 = 3x\ or\ -24 = 3x`

Divide by 3

`12 = x\ or\ -8 = x`

`x =12\ or\ -8`

So, the coordinates are (-8, 11) or (12,11)