Question

1. the hypotenuse of a right isosceles triangle is 5 cm long.

a) write an exact expression for the base and the height of the right
triangle, using primary trigonometric ratios. 4 marks
b) use your expressions to determine the exact area of the triangle.
2 marks

Explanation:

Step1: Find base and height

In a right - isosceles triangle, let the legs (base \(b\) and height \(h\)) be equal. Using Pythagoras' theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse). Since \(a = b\) (right - isosceles) and \(c = 5\), we have \(2a^{2}=25\), so \(a^{2}=\frac{25}{2}\), and \(a=\frac{5}{\sqrt{2}}\). So base \(b=\frac{5}{\sqrt{2}}\) and height \(h = \frac{5}{\sqrt{2}}\).

Step2: Calculate area

The area of a triangle \(A=\frac{1}{2}\times base\times height\). Substitute \(b=\frac{5}{\sqrt{2}}\) and \(h=\frac{5}{\sqrt{2}}\) into the formula: \(A=\frac{1}{2}\times\frac{5}{\sqrt{2}}\times\frac{5}{\sqrt{2}}\). Simplify \(\frac{1}{2}\times\frac{25}{2}=\frac{25}{4}\) \(cm^{2}\).

Answer:

The area of the right - isosceles triangle is \(\frac{25}{4}\) \(cm^{2}\).