Find P and calculate PR.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for questions 3.5.1 and 3.5.2.

Identify the known angles in $\triangle PQR$ .
The triangle is a right-angled triangle, so $\angle Q = 90^\circ$ .
We are given $\angle R = 30^\circ$ .
Use the property that the sum of angles in a triangle is $180^\circ$ .
$\angle P + \angle Q + \angle R = 180^\circ$
Substitute the known values:
$\angle P + 90^\circ + 30^\circ = 180^\circ$
Solve for $\angle P$ .
$\angle P + 120^\circ = 180^\circ$ $\angle P = 180^\circ - 120^\circ$ $\angle P = 60^\circ$
The measure of angle P is $\boxed{\text{}60^\circ\text{}}$

3.5.2 Calculate PR.
Identify the known side and angle relative to the side we want to find.
We know the side opposite to $\angle R$ is $PQ = 70 \text{ cm}$ .
We know $\angle R = 30^\circ$ .
We want to find the hypotenuse, PR.
Choose the appropriate trigonometric ratio.
The sine function relates the opposite side and the hypotenuse:
$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
In $\triangle PQR$ , for $\angle R$ :
$\sin(R) = \frac{PQ}{PR}$
Substitute the known values into the equation.
$\sin(30^\circ) = \frac{70 \text{ cm}}{PR}$
Solve for PR.
We know that $\sin(30^\circ) = 0.5$ .
$0.5 = \frac{70 \text{ cm}}{PR}$ $PR = \frac{70 \text{ cm}}{0.5}$ $PR = 140 \text{ cm}$
The length of PR is $\boxed{\text{}140 \text{ cm}\text{}}$

Answer