Question

the circle is inscribed in triangle prt. which statements about the figure are true? select two options. the perimeter of the triangle is 19 units. $overline{tu} cong overline{ts}$ $overline{pu} cong overline{tu}$ the length of line segment pr is 13 units. the length of line segment tr is 10 units.

Explanation:

To solve this, we use the property of a circle inscribed in a triangle (tangent segments from a point to a circle are congruent).

Step 1: Identify Tangent Segments

• From \( T \): \( TU = TS \) (tangents from \( T \) to the circle). Given \( TU = 6 \), so \( TS = 6 \).
• From \( R \): \( RS = RQ \) (tangents from \( R \) to the circle). Given \( RS = 5 \), so \( RQ = 5 \).
• From \( P \): \( PU = PQ \) (tangents from \( P \) to the circle). Given \( PU = 8 \), so \( PQ = 8 \).

Analyze Each Option:

1. Perimeter of \( \triangle PRT \):

Sides: \( PT = PU + TU = 8 + 6 = 14 \), \( TR = TS + RS = 6 + 5 = 11 \), \( PR = PQ + RQ = 8 + 5 = 13 \).
Perimeter: \( 14 + 11 + 13 = 38 \). So "perimeter is 19" is false.

1. \( \overline{TU} \cong \overline{TS} \):

By the tangent segment property, \( TU = TS \) (both 6). So this is true.

1. \( \overline{PU} \cong \overline{TU} \):

\( PU = 8 \), \( TU = 6 \). Not congruent. False.

1. Length of \( PR \):

\( PR = PQ + RQ = 8 + 5 = 13 \). True.

1. Length of \( TR \):

\( TR = TS + RS = 6 + 5 = 11 \), not 10. False.

Answer:

B. \( \overline{TU} \cong \overline{TS} \)
D. The length of line segment \( PR \) is 13 units.