Question

Point P is the incenter of ΔRST. Point P is the incenter of triangle R S T. Lines are drawn from each point of the triangle to point P. Lines are drawn from point P to the sides of the triangle to form right angles and line segments P A, P B, and P C. The triangle has different angle measures. Which must be true? A. begin mathsize 14px style top enclose PC space approximately equal to space top enclose PR end style B. begin mathsize 14px style top enclose PA space approximately equal to space top enclose PB end style C. begin mathsize 14px style angle RSP space approximately equal to space angle RTP end style D. begin mathsize 14px style angle SPB space approximately equal to space angle TPA end style

Answer 1

The correct statement is B: PA≈PB.

To determine which statement is true, let's consider the properties of an incenter in a triangle.

The incenter of a triangle is the point where the angle bisectors meet. The angle bisectors divide each angle of the triangle into two equal parts.

In the given scenario:

Point P is the incenter of triangle RST.

Lines are drawn from each vertex of the triangle to point P.

Lines are drawn from point P to the sides of the triangle to form right angles.

Now, let's analyze the given options:

A.PC≈PR: This is not necessarily true. The incenter does not create congruent segments when lines are drawn to the sides of the triangle.

B. PA≈PB: This is true. The incenter is equidistant from the sides of the triangle, so the segments PA, PB, and PC are approximately equal.

C. ∠RSP≈∠RTP:

This is not necessarily true. The angles formed by the incenter and the vertices of the triangle are not necessarily congruent.

D. ∠SPB≈∠TPA:

This is not necessarily true. The angles formed by the incenter and the sides of the triangle are not necessarily congruent.

Therefore, the correct statement is B: PA≈PB.