Question

An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from `(7 ,5 )` to `(8 ,2 )` and the triangle's area is `36`, what are the possible coordinates of the triangle's third corner?

Answer 1

Coordinates of the third corner `color(purple)(A = (1452/211), (695/211))`

Explanation:

Refer diagram.

`a = sqrt((8-7)^2 + (2-5)^2) = 3.1623`

Midpoint of base BC = `(8+7)/2, (2+5)/2 = (15/2, 7/2)`

Slope of BC `m_(BC) = (2-5) / (8-7) = -3`

Slope of altitude AD of the triangle passing through the midpoint D is `m_(AD) = -1/(m_(BC)) = -(1/-3) = 1/3`

Equation of altitude AD is

`y - (7/2) = (1/3) * (x - (15/2))`

`6y - 21 = 2x - 15`

`3y - x = 3`. `color(blue)(Eqn) color(white)(xx)color (blue)(1)`

Area of triangle `A = (1 / 2) a h`

`h = (2 * 36) / 3.1623 = 22.7682`

Slope of AB = `tan theta = m_(AB) = h / (a/2) = 22.7682 / (3.1623/2) = 14.4`

Equation of AB is

`y - 5 = 14.4 * (x - 7)`

`y - 14.4x = -95.8`. `color(blue)(Eqn) color(white)(xx)color (blue)(2)`

Solving Eqns (1) & (2) we get the coordinates of vertex ‘A’

`color(purple)( A (1452/211), (695/211))`

Verification :

Length of `AB = b = sqrt((7 - (1452/211))^2 + (5 - (695/211)^2) = 1.7103`

Length of `AC = b = sqrt((8-(1452/211))^2 + (2 - (695/211)’^2) = 1.7103`

Value of side `AB = AC = 1.7103`.

Hence its an isosceles triangle with sides `color(green)(3.1623, 1.7103, 1.7103)`