Question

Two corners of an isosceles triangle are at `(8 ,7 )` and `(2 ,3 )`. If the triangle's area is `64`, what are the lengths of the triangle's sides?

Answer 1

See a solution process below:

Explanation:

The formula for the area of an isosceles triangle is:

`A = (bh_b)/2`

First, we must determine the length of the triangles base. We can do this by calculating the distance between the two points given in the problem. The formula for calculating the distance between two points is:

`d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)`

Substituting the values from the points in the problem gives:

`d = sqrt((color(red)(2) - color(blue)(8))^2 + (color(red)(3) - color(blue)(7))^2)`

`d = sqrt((-6)^2 + (-4)^2)`

`d = sqrt(36 + 16)`

`d = sqrt(52)`

`d = sqrt(4 xx 13)`

`d = sqrt(4)sqrt(13)`

`d = 2sqrt(13)`

The Base of the Triangle is: `2sqrt(13)`

We are given the area is `64`. We can substitute our calculation above for `b` and solve for `h_b`:

`64 = (2sqrt(13) xx h_b)/2`

`64 = sqrt(13)h_b`

`64/color(red)(sqrt(13)) = (sqrt(13)h_b)/color(red)(sqrt(13))`

`64/sqrt(13) = (color(red)(cancel(color(black)(sqrt(13))))h_b)/cancel(color(red)(sqrt(13)))`

`h_b = 64/sqrt(13)`

The Height of the Triangle is: `64/sqrt(13)`

To find the length of the triangles sides we need to remember the mid-line of an isosceles:
- bisects the base of the triangle into two equal parts
- forms a right angle with the base

Therefore, we can use the Pythagorean Theorem to find the length of the side of the triangle where the side is the hypotenuse and the height and `1/2` the base are the sides.

`c^2 = a^2 + b^2` becomes:

`c^2 = (1/2 xx 2sqrt(13))^2 + (64/sqrt(13))^2`

`c^2 = (sqrt(13))^2 + (64/sqrt(13))^2`

`c^2 = 13 + 4096/13`

`c^2 = 169/13 + 4096/13`

`c^2 = 4265/13`

`sqrt(c^2) = sqrt(4265/13)`

`c^2 = (sqrt(25)sqrt(185))/sqrt(13)`

`c^2 = (5sqrt(185))/sqrt(13)`

The Length of the Triangle's Side is: `(5sqrt(185))/sqrt(13)`