Question

The longest side of a right-angled triangle is 13cm. If the perimeter of the triangle is 30cm, find the shortest side.

Answer 1

There is a quick solution and a long solution. The quick arises from recognizing that this is a right triangle where the longest side (A.K.A. the hypotenuse) has length 13. Recall that (5, 12, 13) Pythagorean triple, and that 5+12+13=30. That means the other two sides must have length 12 and 5. The shorter side is 5.

The longer way involves more practice with algebra and can be reviewed below:

As the longest side, 13 cm must be length of the hypotenuse. Let a be the length of one leg, and b the length of the other leg. Then the Pythagorean theorem says that

\[a^2 +b^2= 13^2=169.\]

Moreover, since the perimeter of this triangle is 30, we also know that (a+b+c)=30, or equivalently (substitute c=13 and solve for a) we have a=17-b.

Now use the fact that a=17-b to simplify the Pythagorean theorem:

\[a^2 +b^2 =169\]

\[(17-b)^2 +b^2=169\ now expand (17-b)^2

\[289 - 34b +b^2 + b^2=169\] Bring 169 to the other side and factor

\[120 - 34b +2b^2=0 \ \ \ \ \ \text{ or } \ \ \ 2(60-17b+b^2)=0\] This is equivalent to solving

\[60-17b+b^2=0.\] Now use the quadratic formula:

b=\[\frac{17\pm \sqrt{289-240}}{2}\]=\[\frac{17\pm 7}{2}\] = 12 or5.

These represent the lengths of the two other sides of this triangle; the shorter side is of course 5. Your answer is 5.