Finding perimeter knowing area of right angle triangle and ratio of sides

Question

Hi I am stuck on this question (GCSE extended maths) I can sort of get to the correct answer but I am not clear why. There is no clear explanation (or at all) how to do this in any texts I have looked at. The question looks like a curve ball to me, but I am sure I am missing something:

The area of a right-angled triangle is $51.2cm^{2}.$

If the two perpendicular sides are in the ratio $1:1.6,$ find the lengths of all three sides of the triangle.

Answer 1

Let $a^2+b^2=c^2$ then

• $\frac12 ab = 51.2$
• $\frac{a}b=\frac{1}{1.6}$

$$\implies b=1.6 a \implies0.8a^2=51.2\implies a=8 \quad b=12.8 \quad c=\sqrt{8^2+12.8^2}$$

Answer 2

The ratio of sides is 1:1.6

or lengths $x$ and $1.6x$

Area $= \frac 12 bh\\ 0.8 x^2 = 51.2\\x^2=64\\x = 8$

Two of the sides then are $8$ and $12.8$

The 3rd side you can find using the Pythagorean theorem.