   Chapter 6.4, Problem 24E 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

Quadrilateral R S T V is inscribed in circle Q (not shown). If arcs R S ⌢ , S T ⌢ , and T V ⌢ are all congruent, what type of quadrilateral is R S T V ?

To determine

To find:

The type of quadrilateral is RSTV.

Explanation

Given:

Quadrilateral RSTV which is inscribed in circle Q and arcs RS, ST, and TV are all congruent.

Property Used:

The measure of an inscribed angle of a circle is half the measure of its intercepted arc. Consider the figure below, let the inscribed angle be 2

Then m2=12mHJ and generally HKKJ.

If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel.

A trapezoid is a special quadrilateral that has only one pair of parallel sides. An isosceles trapezoid has one pair of parallel sides and the base angles are congruent.

Calculation:

Suppose RSTV is a quadrilateral inscribed in circle Q and arcs RS, ST, and TV are all congruent

Since, the measure of an inscribed angle of a circle is half the measure of its intercepted arc therefore, mS=12mTVR, mV=12mRST, mT=12mVRS and mR=12mSTV.

Calculating the sum of interior angles on the same side mS+mR,

mS+mR=12mTVR+12mSTV

mS+mR=12(mTV+mVR)+12(mST+mTV)

Also, RS, ST, and TV are all congruent. Substituting RS=ST=TV,

mS+mR=12mTV+12mVR+12mTV+12mTV

mS+mR=12(3mTV+mVR)

Also, in a circle,

mTV+mVR+mRS+mST=360°

Since, RS, ST, and TV are all congruent

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