Exercise 43 Page 25

Practice makes perfect

a We are given the following diagram and asked to create an algebraic expression that represents GK.

Normally we could use the Segment Addition Postulate to find the segment length. However, when adding GJ and HK, we have an overlap of the segment length HJ. We can subtract the value of HJ one time so that it is not doubled. This gives us the following equation. GK= GJ+ HK- HJ We can substitute the expressions and simplify.

GJ+HK-HJ

SubstituteIII

Substitute GJ= 2x+3, HK= 4x-3, HJ= x

2x+3+ 4x-3- x

AddSubTerms

Add and subtract terms

We now have that the algebraic expression 5x represents GK.

b Now we want to find the values of GH and JK given that GK=30. From Part A, we know that GK=5x, so we can substitute 30 for GK and solve for the x.

GK=5x

Substitute

GK= 30

30=5x

DivEqn

.LHS /5.=.RHS /5.

6=x

RearrangeEqn

Rearrange equation

x=6

Now that we have the value of x, we can substitute it in to the expressions and find the values of HK, GJ, and HJ.

Segment	Expression	x=6	Length
HK	4x-3	4( 6)-3	21
GJ	2x+3	2( 6)+3	15
HJ	x	6	6

Now we have the following values on our diagram.

Notice that GH and JK, the lengths we want to find, are both not given on the graph. We need to find them using the Segment Addition Postulate. Let's find GH first by utilizing the relationship between GH, HJ, and GJ.

GH+HJ=GJ

SubstituteII

GJ= 15, HJ= 6

GH+ 6= 15

SubEqn

LHS-6=RHS-6

GH=9

Now we can follow a similar process to find the length JK.

HJ+JK=HK

SubstituteII

HK= 21, HJ= 6

6+JK= 21