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.

1) The reference triangle has perimeter 4+3+5 = 12. The middle side is (12 cm)/4 = 3 cm times the length of that of the reference triangle. The perimeter has the same ratio, so will be
.. P = 12*(3 cm) = 36 cm . . . . . . perimeter

2) The actual triangle has legs 9 cm and 12 cm. The area is half their product,
.. A = (1/2)*(12 cm)*(9 cm) = 54 cm^2 . . . . . . area

3) The height satisfies the equation
.. A = (1/2)bh
where b is the hypotenuse length and h is the height of interest. Filling in the numbers, we have
.. 54 cm^2 = (1/2)(15 cm)*h . . . . . hypotenuse = (3 cm)*5 = 15 cm
.. h = (108/15) cm = 7.2 cm . . . . . height to the hypotenuse

The mode of the numbers is 29.