In a triangle, the lengths of two heights are 8, and the third is 12. Find the area of the triangle.

Since a triangle has two heights, this triangle is isosceles.

AB = CB.

Then the height of the HВ will divide the base of the speaker in half. AH = CH = AC / 2.

The area of the triangle ABC is equal to:

S = AC * BH / 2 = AC * 6 cm2.

S = AB * CM / 2 = AB * 4 cm2.

Then AC * 6 = AB * 4.

AC = AB * 4/6 = AB * 2/3.

Then AH = AC / 2 = (AB * 2/3) / 2 = AB / 3.

From the right-angled triangle ABН we express AB.

AB ^ 2 = BH ^ 2 + AH ^ 2 = 144 + (AB / 3) ^ 2 = 144 + AB ^ 2/9.

8 * AB ^ 2/9 = 144.

AB ^ 2 = 144 * 9/8 = 162.

AB = 9 * √2 cm.

Determine the area of the triangle.

S = AB * CM / 2 = 9 * √2 * 8/2 = 36 * √2 cm2.

Answer: The area of the triangle is 36 * √2 cm2.