Question

find the missing side lengths. leave your answers as radicals in simplest form.
image of a right triangle with one acute angle 45°, one leg 5, hypotenuse a, and the other leg b
d) $a = \frac{5\sqrt{6}}{3}, b = \frac{5\sqrt{3}}{2}$
b) $a = 5\sqrt{2}, b = \frac{5\sqrt{3}}{2}$
a) $a = 5\sqrt{2}, b = 5$
c) $a = 5, b = 5\sqrt{2}$
rewatch

Explanation:

Step1: Identify Triangle Type

The triangle is a right - angled triangle with one angle equal to \(45^{\circ}\). So, the third angle is also \(45^{\circ}\) (since the sum of angles in a triangle is \(180^{\circ}\), and \(180 - 90 - 45=45^{\circ}\)). This is a \(45 - 45 - 90\) triangle. In a \(45 - 45 - 90\) triangle, the legs are equal in length, and the hypotenuse \(h\) is related to the leg length \(l\) by the formula \(h = l\sqrt{2}\).

Step2: Determine Leg and Hypotenuse

One of the legs (the horizontal side) has a length of \(5\). Since the triangle is \(45 - 45 - 90\), the other leg \(b\) (the vertical side) is also equal to \(5\) (because the two legs opposite the \(45^{\circ}\) angles are equal). The hypotenuse \(a\) can be found using the formula for the hypotenuse of a \(45 - 45 - 90\) triangle. If the leg length \(l = 5\), then the hypotenuse \(a=l\sqrt{2}=5\sqrt{2}\).

Answer:

A) \(a = 5\sqrt{2},b = 5\)