Question

special right triangles
consider triangle pqr. what is the length of side qr?
options: 8 units, 16 units, 8√3 units, 16√3 units
(image of right triangle pqr with right angle at p, pq = 8√3, pr = 8)

Explanation:

Step1: Identify the triangle type

Triangle \( PQR \) is a right - triangle with \( \angle P = 90^{\circ} \), \( PR = 8 \) and \( PQ=8\sqrt{3} \). We can also check the angles. For a right - triangle, if the sides are in the ratio \( 1:\sqrt{3}:2 \), it is a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle. Here, the shorter leg \( PR = 8 \), the longer leg \( PQ = 8\sqrt{3} \), so the hypotenuse \( QR \) should be twice the shorter leg.

Step2: Apply the Pythagorean theorem (alternative method)

The Pythagorean theorem states that for a right - triangle \( a^{2}+b^{2}=c^{2} \), where \( a \) and \( b \) are the legs and \( c \) is the hypotenuse. Let \( a = 8 \), \( b = 8\sqrt{3} \), then \( c^{2}=8^{2}+(8\sqrt{3})^{2} \)
\[

$$\begin{align*} c^{2}&=64 + 64\times3\\ c^{2}&=64+192\\ c^{2}&=256\\ c&=\sqrt{256}\\ c& = 16 \end{align*}$$

\]

Answer:

16 units (corresponding to the option with "16 units" text)