In ∆ABC with m∠C = 90° the sides satisfy the ratio BC:AC:AB = 4:3:5. If the side with middle length is 12 cm, find: 1) The perimeter of ∆ABC; 2) The area of ∆ABC; 3) The height to the hypotenuse.Please walk me through the question.20 points.

Accepted Solution

First of all, let's find the dimensions of your triangle. The middle length of the ratio is 4, so the scale factor is (12 cm)/4 = 3 cm. Thus the other two side lengths are 3×3 cm = 9 cm, and 5×3 cm = 15 cm.

The area of a triangle is the same no matter how you compute it. If you compute it using the legs of the right triangle, you find it is
  A = (1/2)×(12 cm)×(9 cm) = 54 cm²

If you compute it using the hypotenuse and the height to the hypotenuse, it is the same.
  A = 54 cm² = (1/2)×(15 cm)×height
We solve this for height by dividing by the coefficient of height. Doing so, we find
  height = (54 cm²)/(7.5 cm) = 7.2 cm

The height to the hypotenuse is 7.2 cm.