Question

find the exact value of each part labeled with a variable in the figure. p = q = r = (simplify your answers, including any radicals. use integers or fractions for any numbers in the expressions.)

Explanation:

Step1: Analyze left - hand right - triangle

In the left - hand right - triangle with angles 45 - 45 - 90 and one side length 45. In a 45 - 45 - 90 triangle, the two legs are equal. Let the legs be \(a\) and \(a\), and the hypotenuse be \(c\). The ratio of the sides is \(a:a:c = 1:1:\sqrt{2}\). Since one leg is 45, the other leg \(p = 45\), and the hypotenuse \(r\) of this 45 - 45 - 90 triangle is \(r=45\sqrt{2}\) using the formula \(c = a\sqrt{2}\).

Step2: Analyze right - hand right - triangle

In the right - hand right - triangle with a 30 - 60 - 90 angle - set and hypotenuse \(r = 45\sqrt{2}\). The ratio of the sides of a 30 - 60 - 90 triangle is \(a:b:c=1:\sqrt{3}:2\), where \(a\) is the side opposite the 30 - degree angle, \(b\) is the side opposite the 60 - degree angle, and \(c\) is the hypotenuse. If \(c = 45\sqrt{2}\), and the side opposite the 30 - degree angle \(q\), then \(q=\frac{r}{2}\) (because in a 30 - 60 - 90 triangle, the side opposite the 30 - degree angle is half of the hypotenuse). So \(q=\frac{45\sqrt{2}}{2}\).

Answer:

\(p = 45\), \(q=\frac{45\sqrt{2}}{2}\), \(r = 45\sqrt{2}\)