Question

1. which of the following statements is true?

figure 3 shows a triangle \\(pqr\\) with a line segment \\(rs\\) perpendicular to \\(pq\\) at point \\(s\\). the length of \\(sq\\) is \\(10\\). the angle \\(\angle spr = 60^\circ\\) and the angle \\(\angle srq = 30^\circ\\).

• \\(sr = 10\sqrt{3}\\)
• \\(ps = 10\sqrt{3}\\)
• \\(pr = 10\\)
• \\(pq = 10\\)

Explanation:

🆕 New Concept Discovered: Special Right Triangles (30-60-90)
Ratios of sides in a 30-60-90 triangle

Step 1: Analyze the right triangle RSQ

The line segment \(RS\) is perpendicular to \(PQ\), which creates two right triangles: \(\triangle PSR\) and \(\triangle QSR\).

Looking at the right triangle \(\triangle QSR\):

• The angle at \(S\) is \(90^\circ\).
• The angle at \(R\) (\(\angle SRQ\)) is given as \(30^\circ\).
• Since the angles in a triangle add up to \(180^\circ\), the angle at \(Q\) (\(\angle SQR\)) must be:

$$ 180^\circ - 90^\circ - 30^\circ = 60^\circ $$

This makes \(\triangle QSR\) a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) special right triangle.

Step 2: Use 30-60-90 triangle ratios to find SR

In any \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the lengths of the sides opposite to the angles \(30^\circ\), \(60^\circ\), and \(90^\circ\) are in the ratio:

$$ 1 : \sqrt{3} : 2 $$

For \(\triangle QSR\):

• The side opposite the \(30^\circ\) angle is \(SQ = 10\).
• The side opposite the \(60^\circ\) angle is \(SR\).

Using the ratio, the side opposite the \(60^\circ\) angle is \(\sqrt{3}\) times the side opposite the \(30^\circ\) angle:

$$ SR = SQ \cdot \sqrt{3} = 10\sqrt{3} $$

This matches the first option: \(SR = 10\sqrt{3}\).

Answer:

\(SR = 10\sqrt{3}\)