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.

*The side with the middle length is 12cm, therefore BC must be 12 cm


Ratio 

BC:AC:AB
  4 : 3 : 5


Find 1 share

Given that BC = 12 cm
4 shares = 12 cm
1 share = 12 ÷ 4 = 3cm


Find AC (4 shares)

3 shares = 3 x 3 = 9 cm
BC = 9cm


Find AB (5 shares)

5 shares = 5 x 3 = 15cm
AB = 15cm


Length

   BC   :  AC   :  AB
12cm  :  9cm : 15cm


Perimeter

12 + 9 + 15 = 36cm


The perimeter is 36cm



Area

1/2 x 9 x 12 = 54cm²


Area = 54cm²


Given that the biggest ratio is AB
⇒AB is the hypotenuse
AB = 15 cm


Hypotenuse = AB = 15cm

The Hypotenuse AB = 20cm

Hope I helped!

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~ Zoe