Question

select all the correct answers. what is the length of side pq in this figure? a. 4 b. 14 c. $sqrt{32}$ d. $sqrt{61}$

Explanation:

1. First, use the Pythagorean - theorem:

• In right - triangle \( \triangle PQR \), we can consider two right - triangles formed. Let's first find the length of the horizontal segment from the right - angle vertex to point \( Q \) using the right - triangle with sides \( 3 \) and \( 5 \).
• By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), in the right - triangle with legs \(3\) and the horizontal leg (let's call it \(x\)) and hypotenuse \(5\), we have \(x=\sqrt{5^{2}-3^{2}}=\sqrt{25 - 9}=\sqrt{16}=4\).
• Now, in the right - triangle with legs \(6\) and \(4\) (the vertical leg is \(6\) and the horizontal leg is the \(x = 4\) we just found) and hypotenuse \(PQ\).
• Again, using the Pythagorean theorem \(PQ=\sqrt{6^{2}+4^{2}}=\sqrt{36 + 16}=\sqrt{52}\).

1. Identify the discipline and sub - field:

• The problem is about finding the length of a side in a right - triangle, which falls under the discipline of Mathematics, specifically the sub - field of Geometry.

Answer:

C. \(\sqrt{52}\)