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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 Review Exercises 21 to 30, use the drawings, where provided, to solve each problem, Angle measures should be found to the nearest degree; lengths should be found to the nearest tenth of a unit.

The diagonals of a rhombus measure 12 in. and 16 in. Find the measure of the obtuse angle of the rhombus.

To determine

To find:

The measure of the obtuse angle of the rhombus such that the rhombus measure12 in and 16 in.

Explanation

Procedure used:

In any triangle ABC such that

AB=c,

BC=a,

CA=b,

mA=α,

mB=β, and

mC=γ.

The Law of cosines is given by

a2=b2+c2-2bccosα

b2=c2+a2-2cacosβ

c2=a2+b2-2abcosγ

The Law of sines is given by

asinα=bsinβ=csinγ

Calculation:

Given:

The length of the diagonals of the rhombus are 12 in and 16 in. Let ABCD be the rhombus such that AC and BD are the two diagonals intersect at ‘O’.

Here,

BD=12 in., and

AC=16 in.

By the property if rhombus, the diagonals are the perpendicular bisector of itself.

Thus,

OB=OD=12BD

OB=OD=1212

OB=OD=6 in.

Also,

OA=OC=12AC

OA=OC=1216

OA=OC=8 in.

Assume that the length of each of the side of the rhombus is ‘x’ in.

Consider the triangle AOD of the rhombus.

OA=8 in,

OD=6 in,

AD=x, and

mAOD=90°

Apply Pythagorean theorem in the AOD.

AD2=OA2=OD2

x2=82+62

x2=64+36

x2=100

Taking square root,

x= 10 in

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