Folding in half
Problem
The shorter sides of a right-angled isosceles triangle are each $10$cm long.
The triangle is folded in half along its line of symmetry to form a smaller triangle.
How much longer is the perimeter of the larger triangle than that of the smaller?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Since the original triangle is isosceles and right-angled, folding it produces a smaller triangle, also isosceles and right-angled.
By Pythagoras' Theorem, the hypotenuse of the original triangle is $\sqrt{200}=10\sqrt{2}$ cm.
Hence the difference between the perimeters of the two triangles is $(10+10+10\sqrt{2})-(5\sqrt{2}+5\sqrt{2} +10)=10$ cm.
Alternatively: let the length of the shorter sides of the new triangle be x cm, shown below. Then the perimeter of the original triangle is $(20+2x)$ cm and the perimeter of the new triangle is $(10+2x)$cm. Hence the difference between the perimeters of the two triangles is $10$cm.
