   Chapter 7.2, Problem 29E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

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

Given: Isosceles Δ R S T R S = R T = 17         a n d         S T = 16. Medians R Z ¯ , T X ¯ and S Y ¯ meet at centroid Q .Find: R Q         and           S Q Exercises 29, 30

To determine

To find:

The RQandSQ in the given isosceles triangle RST.

Explanation

Definition:

The three medians of a triangle are congruent at a point that is two-thirds the distance from any vertex to the midpoint of the opposite side.

Formula used:

The triangle RST, with sides a,bandc opposite vertices R,SandT. Then the length of the median ma form R is given by

ma=122(b2+c2)a2

The length of the median mb form S is given by

mb=122(c2+a2)b2

Where ma,mb denotes the length of the median

Given:

In ΔRST, medians RZ¯, TX¯ and SY¯ intersect at centroid Q. If RS=RT=17andST=16.

Calculation:

The point of concurrence Q for the three medians of a triangle is known as centroid of the triangle. In ΔMNP, medians MB¯, NA¯ and PC¯ intersect at centroid Q.

According to the definition,

RQ=23(RZ)(1)SQ=23(SY)(2)

Given RS=RT=17andST=16.

The triangle RST, with sides a,bandc opposite vertices R,SandT.

So, ST=16=a,RT=17=bandRS=17=c.

We sketch the figure of the isosceles triangle RST with sides a,bandc

First we need to find length of the median RZ,

Using the above formula,

The length of the median ma form R is given by

RZ=ma=122(b2+c2)a2=122(172+172)162=122(289+289)256=122(578)256=121156256=12900

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