6. the hypotenuse of a right isosceles triangle is 5 cm long.\na) write an exact expression for the base and…

Explanation:

Step1: Find base and height using trigonometric ratios

In a right - isosceles triangle, the two non - right angles are (45^{\circ}). Let the hypotenuse (c = 5) cm. Using the sine ratio (\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}) and cosine ratio (\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}). For (\theta = 45^{\circ}), (\sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}). If we let the base (b) and height (h) (since it is isosceles, (b = h)), and using (\sin45^{\circ}=\frac{h}{c}) (or (\cos45^{\circ}=\frac{b}{c})) (h = b=5\times\sin45^{\circ}=5\times\frac{\sqrt{2}}{2})

Step2: Calculate the area of the triangle

The area of a triangle (A=\frac{1}{2}\times b\times h). Since (b = h=\frac{5\sqrt{2}}{2}), then (A=\frac{1}{2}\times\frac{5\sqrt{2}}{2}\times\frac{5\sqrt{2}}{2}) [ \begin{align*} A&=\frac{1}{2}\times\frac{25\times2}{4}\ &=\frac{25}{4} \end{align*} ]

Answer:

a) The base (b = \frac{5\sqrt{2}}{2}) cm and the height (h=\frac{5\sqrt{2}}{2}) cm. b) The area of the triangle (A = \frac{25}{4}\text{ cm}^2)