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
A:
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