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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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BuyFindarrow_forward

Elementary Geometry For College St...

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

In Exercises 46 to 49, prove the stated theorem.

If a line through the center of a circle bisects a chord other than a diameter, then it is perpendicular to the chord. See Figure 6.38 on page 298.

To determine

To find:

To prove the following stated theorem,

“If a line through the center of a circle bisects a chord other than a diameter, then it is perpendicular to the chord”.

Explanation

Given that circle O; OM is the bisector of chord RS¯.

To prove OMRS¯

The diagrammatic representation is given below,

Consider the triangles ΔROMandΔSOM. We know that OR¯=OS¯ since OR and OS are the radius of the circle ( radius is same for all points on the circle).

Then OM=OM. Since it is common for both triangles.

Given that OM is the bisector of chord RS¯ therefore,

RM¯=MS¯

By using the SSS postulate (the three sides of one triangle is equal to the corresponding three sides of the another then thses two triangles are congruent).

Therefore, ΔROMΔSOM

By using CPCTC (corresponding parts of congruent triangles are congruent) to get the following,

mRMO=mSMO

In line RS¯, the mRMOandmSMO form a line pair therefore to get the following,

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