Two corners of an isosceles triangle are at #(2 ,4 )# and #(3 ,8 )#. If the triangle's area is #64 #, what are the lengths of the triangle's sides?

To find the lengths of the sides of the isosceles triangle, we first need to calculate the length of the base, which is the side opposite the vertex angle. The formula for the area of a triangle is given by (base * height) / 2. Since we know the area and the height, we can rearrange the formula to solve for the base length. Then, we can use the distance formula to find the lengths of the other two sides, which are congruent in an isosceles triangle.

Given:

• Area = 64
• Height = distance between the vertex and the line containing the base

Using the formula for the area of a triangle: Area = (base * height) / 2

64 = (base * height) / 2

We know the height is the perpendicular distance from the vertex to the line containing the base. To find this height, we need to find the equation of the line containing the base. Since the triangle is isosceles, the midpoint of the base will be equidistant from the two given vertices. We can find the midpoint by averaging the x-coordinates and the y-coordinates of the given points.

Midpoint coordinates: x-coordinate: (2 + 3) / 2 = 2.5 y-coordinate: (4 + 8) / 2 = 6

Now, we have the midpoint of the base as (2.5, 6). Using this midpoint and one of the given vertices (let's say (2, 4)), we can find the equation of the line containing the base. With the equation of the base, we can find the height by calculating the perpendicular distance from the vertex (3, 8) to this line.

After finding the height, we can solve for the base length. Once we have the base length, we can use the distance formula between the given vertices and the midpoint of the base to find the lengths of the congruent sides.