Question

here is triangle pqr.
find the length of each leg.
type your answers in the boxes.
pq = \boxed{} units
qr = \boxed{} units

Explanation:

Step1: Identify triangle type

Triangle \( PQR \) is a right - angled triangle (\( \angle Q = 90^{\circ} \)) with \( \angle R=30^{\circ} \) and \( \angle P = 60^{\circ} \), and hypotenuse \( PR = 4 \) units. In a 30 - 60 - 90 triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), where the side opposite \( 30^{\circ} \) is the shortest leg, the side opposite \( 60^{\circ} \) is the longer leg, and the hypotenuse is twice the shortest leg.

Step2: Find length of \( PQ \) (opposite \( 30^{\circ} \))

The side \( PQ \) is opposite \( \angle R = 30^{\circ} \). In a 30 - 60 - 90 triangle, the length of the side opposite \( 30^{\circ} \) is half of the hypotenuse. So, \( PQ=\frac{1}{2}\times PR \). Since \( PR = 4 \), then \( PQ=\frac{1}{2}\times4 = 2 \) units.

Step3: Find length of \( QR \) (opposite \( 60^{\circ} \))

The side \( QR \) is opposite \( \angle P=60^{\circ} \). In a 30 - 60 - 90 triangle, the length of the side opposite \( 60^{\circ} \) is \( \sqrt{3} \) times the length of the side opposite \( 30^{\circ} \). We know the side opposite \( 30^{\circ} \) ( \( PQ \)) is 2 units. So, \( QR=\sqrt{3}\times PQ \). Substituting \( PQ = 2 \), we get \( QR = 2\sqrt{3} \) units. Also, we can use the Pythagorean theorem: \( QR=\sqrt{PR^{2}-PQ^{2}}=\sqrt{4^{2}-2^{2}}=\sqrt{16 - 4}=\sqrt{12}=2\sqrt{3} \) units.

Answer:

\( PQ = \boldsymbol{2} \) units, \( QR=\boldsymbol{2\sqrt{3}} \) units