Exercise 75 Page 530

Let's add the given side lengths to the diagram.

In any parallelogram, the diagonals bisect each other. Since a rhombus is a parallelogram, we know that DM≅ MG. Let's add this information to the diagram.

Therefore, we can set the expressions of DM and MG equal to each other. 4x-3= x+6 Let's solve this equation by calculating x.

4x-3=x+6

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Solve for x

SubEqn

LHS-x=RHS-x

3x-3=6

AddEqn

LHS+3=RHS+3

3x=9

DivEqn

.LHS /3.=.RHS /3.

x=3

To determine DG, we can use the Segment Addition Postulate.

Segment Addition Postulate

If A, B and C are collinear, then Point B is between A and C if and only if AB+BC=AC.

By setting DG, equal to the sum of DM and MG and then substituting x=3 into the simplified expression, we can calculate DG.

DG=DM+MG

SubstituteII

DM= 4x-3, MG= x+6

DG=( 4x-3)+( x+6)

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Simplify right-hand side

RemovePar

Remove parentheses

DG=4x-3+x+6

AddTerms

Add terms

DG=5x+3

Substitute

x= 3

DG=5( 3)+3

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Evaluate right-hand side

Multiply

Multiply

DG=15+3

AddTerms

Add terms

DG=18