Question

if ∠rsu ≅ ∠tsu and tu = 96, what is ru? diagram: u connected to t (right angle at t), u connected to r (right angle at r), s connected to t, s connected to r, s connected to u. ru =

Explanation:

Step1: Identify the Angle Bisector and Right Triangles

We know that \( \angle RSU \cong \angle TSU \), so \( SU \) is the angle bisector of \( \angle RST \). Also, \( \angle T = 90^\circ \) and \( \angle R = 90^\circ \), meaning \( UT \perp ST \) and \( UR \perp SR \).

Step2: Apply the Angle Bisector Theorem for Distances

By the Angle Bisector Theorem, a point on the angle bisector is equidistant from the two sides of the angle. Since \( SU \) bisects \( \angle RST \), and \( U \) is on \( SU \), the distance from \( U \) to \( ST \) (which is \( TU \)) and the distance from \( U \) to \( SR \) (which is \( RU \)) should be equal.

Step3: Set the Lengths Equal

Given \( TU = 96 \), and from the angle bisector property, \( RU = TU \).

Answer:

\( 96 \)