Question

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.

Answer 1

Hypotenuse AB = 20cm

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Answer 2

We are given the ratio as BC:AC:AB = 4:3:5

Middle side is AC and its measure is 12 cm.

Total ratio = 4+3+5 = 12

So that means :

$\frac{3x}{12}=12$

x=48

So perimeter =48 cm

Now let us find the other two sides: AB and BC

BC = 4x/ 12 = (4*48)/12= 16 cm

AB = 5x/12 (5*48)/12 =20 cm

Figure shows the actual representation of the triangle.

Area:

Area of triangle is given as :

$Area = \frac{1}{2} * base * height$

here base is 16 cm and height is 12 cm, plugging in formula,

$Area = \frac{1}{2} * 16 * 12$

Area = 96 square cm

Height of Hypotenuse = AB = 20cm