Chapter 5.3, Problem 43E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# The vertices of rhombus ARST lie on Δ A B C as shown. Where A B = c and A C = b , show that R S = b c b + c (Let A R = x .)

To determine

To show:

The side RS=bcb+c by using the given information.

Explanation

Definition:

AA:

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

CSSTP:

Corresponding sides of similar triangles are proportional.

Description:

Given that the vertices of rhombus ARST lie on Î”ABC, where AB=c, AR=x and AC=b.

The given figure is shown below.

Figure

From the given figure, it is observed that vertices of rhombus ARST lie on Î”ABC.

Also it is observed that RBS is a triangle.

It is known that the vertices of rhombus are the same and the opposite sides are parallel.

So, RSÂ =Â x and RSÂ¯âˆ¥ACÂ¯.

If two parallel lines are cut by a transversal corresponding angles are congruent.

That is. âˆ Râ‰…âˆ A.

Now consider two triangles ABC and RBS.

Here, both triangles have the common point B.

Hence, by identity âˆ Bâ‰…âˆ B.

The above mentioned AA definition, Î”ABCâˆ¼Î”RBS since two angles of one triangle are congruent to two angles of another triangle.

From the definition of CSSTP, corresponding sides of similar triangles are proportional.

That is, BRBA=BSBC=RSAC.

Consider first and last proportions to find the values of x

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