Question

$\overline{qr}$ is tangent to circle p at point q.
what is the approximate length of $\overline{rp}$? round to the nearest tenth.
\bigcirc 5.6 units
\bigcirc 6.1 units
\bigcirc 8.3 units
\bigcirc 9.8 units

Explanation:

Step1: Identify the right triangle

Since \( \overline{QR} \) is tangent to circle \( P \) at \( Q \), \( \angle PQR = 90^\circ \) (tangent is perpendicular to radius). So \( \triangle PQR \) is a right triangle with \( PQ = 3 \) (radius), \( QR = 5.3 \), and \( RP \) as the hypotenuse.

Step2: Apply Pythagorean theorem

The Pythagorean theorem states \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Here, \( PQ = 3 \), \( QR = 5.3 \), and \( RP = c \). So:
\[
RP^2 = PQ^2 + QR^2
\]
\[
RP^2 = 3^2 + 5.3^2
\]
\[
RP^2 = 9 + 28.09
\]
\[
RP^2 = 37.09
\]

Step3: Solve for \( RP \)

Take the square root of \( 37.09 \):
\[
RP = \sqrt{37.09} \approx 6.1
\]

Answer:

B. 6.1 units