Question

find the lengths of the missing sides in the triangle. write your answers as integers or as decimals rounded to the nearest tenth.
not drawn to scale
$x = 12.7, y = 9$
$x = 7.8, y = 6.4$
$x = 6.4, y = 7.8$
$x = 9, y = 12.7$

Explanation:

Step1: Identify the triangle type

This is a right - angled triangle with one angle \(45^{\circ}\), so it is a \(45 - 45-90\) triangle. In a \(45 - 45-90\) triangle, the legs are equal. But wait, no, if one of the non - right angles is \(45^{\circ}\), and the other non - right angle is \(180-(90 + 45)=45^{\circ}\). Wait, no, no, actually, if we use trigonometric ratios. Let's assume the side opposite to \(45^{\circ}\) is \(y = 9\) (one of the legs).

Step2: Use trigonometric ratios to find \(x\) (hypotenuse)

We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\theta = 45^{\circ}\), \(\sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}\approx0.707\). If \(y = 9\) (opposite side to \(45^{\circ}\)), and using \(\sin45^{\circ}=\frac{y}{x}\), then \(x=\frac{y}{\sin45^{\circ}}\). Substitute \(y = 9\) into the formula: \(x=\frac{9}{\sin45^{\circ}}=\frac{9}{0.707}\approx12.7\).

Answer:

A. \(x = 12.7,y = 9\)