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. i need to find the height

1- getting the side lengths:
We are given that the ratio between the sides is:
BC : AC : AB
4    : 3    : 5
We are also given that AC = 12 cm
We will simply use cross multiplication to find the lengths of the other two sides as follows:
BC : AC : AB
4    : 3    : 5
??  : 12  : ??
length of BC = (12*4) / 3 = 16 cm
length of AB = (12*5) / 3 = 20 cm

2- getting the perimeter of the triangle:
perimeter = AB + BC + AC
perimeter = 20 + 16 + 12
perimeter = 48 cm

3- getting the length of the hypotenuse:
We are given that angle C is the right angle in triangle ABC. The hypotenuse is the side opposite to the right angle. In our case, this side is AB.
This means that the length of the hypotenuse = AB = 20 cm

4- getting the area of the triangle:
area of triangle = 0.5 * base * height
Since the given triangle is a right-angled triangle, therefore, the base and the height are the two legs associated with the right angle.
This means that:
base = AC = 12 cm
height = BC = 16 cm
Therefore:
area of triangle = 0.5 * 12 * 16 = 96 cm²

Hope this helps :)