Answer:
the area of each triangle is 24 square units.
Explanation:
Since the two triangles have equal areas, we can set their areas equal to each other and solve for the unknown side. Let's call the height of both triangles "h" and the base of the first triangle "b1" and the base of the second triangle "b2". Then we have:
Area of first triangle = 1/2 * b1 * h
Area of second triangle = 1/2 * b2 * h
Since the two areas are equal, we can set them equal to each other:
1/2 * b1 * h = 1/2 * b2 * h
We can cancel the factor of 1/2 and solve for h:
b1 * h = b2 * h
b1 = b2
This means that the bases of the two triangles are equal. Now we can use the Pythagorean theorem to find the height h. Let's call the length of the hypotenuse of the first triangle "c1" and the length of the hypotenuse of the second triangle "c2". Then we have:
c1^2 = b1^2 + h^2
c2^2 = b2^2 + h^2
Since b1 = b2, we can write:
c1^2 = c2^2
Taking the square root of both sides, we get:
c1 = c2
This means that the hypotenuses of the two triangles are equal. So we have two congruent right triangles, and we can use the Pythagorean theorem to find the height h:
c1^2 = b1^2 + h^2
c1^2 - b1^2 = h^2
h = sqrt(c1^2 - b1^2)
We can use the measurements given in the figure to find b1 and c1:
b1 = 8
c1 = 10
Plugging these values into the equation for h, we get:
h = sqrt(10^2 - 8^2) = sqrt(36) = 6
So the height of each triangle is 6, and their areas are:
Area of first triangle = 1/2 * 8 * 6 = 24
Area of second triangle = 1/2 * 10 * 6 = 30
Therefore, the area of each triangle is 24 square units.