   Chapter 3.1, Problem 41E ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

#### Solutions

Chapter
Section ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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# Given: ∠ A B C ; R S ¯ is the perpendicular bisector of A B ¯ ; R T ¯ is theperpendicular bisector of B C ¯ . Prove: A R ¯ ≅ R C ¯ To determine

To prove:

The segment AR¯ is congruent to the segment RC¯.

Explanation

Given:

The given figure is,

Figure (1)

From figure (1).

RS¯ is the perpendicular bisector of AB¯

And,

RT¯ is the perpendicular bisector of BC¯.

Concept:

The perpendicular bisector bisect the segment in two congruent part.

The perpendicular bisector makes an angle of 90° with the bisected segment.

Approach:

Consider the triangle,

ΔRTB and ΔRTC,

Since, RT¯ is the perpendicular bisector. Then,

BT¯TC¯RTBRTC

Since, the side RT¯ is common in both the triangles ΔRTB and ΔRTC. Then,

RT¯RT¯

Thus, from SAS property if two sides and one angle are congruent in two triangles, then

ΔRTBΔRTC

Since, the congruent triangles have congruent corresponding segments. Then,

RB¯RC¯ ……(1)

Consider the triangle,

ΔRSA and ΔRSB,

Since, RS¯ is the perpendicular bisector

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