Right triangles and trigonometry: Practice Questions

In right triangle \(DEF\), the lengths of the legs are \(d = 8\) and \(e = 15\). What is the length of the hypotenuse \(f\)?

In triangle \(PQR\), \(\angle Q\) is a right angle and \(\angle P = 60^\circ\). If the length of side \(PQ\) is 10, what is the length of the hypotenuse \(PR\)?

In a right triangle, one of the acute angles is \(x^\circ\). If \(\sin(x^\circ) = \frac{5}{13}\), what is the value of \(\cos(90^\circ - x^\circ)\)?

In right triangle \(ABC\), \(\angle C\) is a right angle. If the length of side \(AC\) is 3 and the length of the hypotenuse \(AB\) is 5, what is the value of \(\sin B\)?

A 13-foot ladder is leaning against a vertical wall. The base of the ladder is 5 feet away from the bottom of the wall. How many feet above the ground is the top of the ladder?

In right triangle \(LMN\), angle \(M\) is a right angle and \(\tan L = \frac{3}{4}\). If the length of the side adjacent to angle \(L\) is \(16\), what is the length of the side opposite to angle \(L\)?

In right triangle \(PQR\), the measure of \(\angle Q\) is \(90^{\circ}\) and the measure of \(\angle P\) is \(30^{\circ}\). If the length of the hypotenuse \(PR\) is \(12\sqrt{3}\), what is the length of side \(QR\)?

In right triangle \( ABC \), the measure of \( ∠C \) is \( 90^∘ \) and the measure of \( ∠A \) is \( 30^∘ \). The length of the hypotenuse \( AB \) is \( 12 \) units. Point \( D \) lies on \( AC \) such that \( ∠BDC = 45^∘ \).

Part A: Determine the lengths of sides \( BC \) and \( AC \) using the properties of a \( 30-60-90 \) triangle.
Part B: Find the length of segment \( CD \) using the properties of the \( 45-45-90 \) triangle \( BDC \).
Part C: Calculate the length of segment \( AD \).