Exercise 12 Page 315

We are told that N is the incenter of the triangle. This means that it is the point of concurrency of the angle bisectors.

Note that the segments connecting N with the sides, NG, NH, and NJ are perpendicular to the sides. This means that their lengths are the distance from each side to the incenter. According to the Incenter Theorem, the incenter of a triangle is equidistant from its sides. Therefore, their lengths are equal. NG=NH=NJ Since we are given the expression for the lengths of NG and NH, we can equate them to solve for x.

NG=NH

SubstituteII

NG= x+3, NH= 2x-3

x+3= 2x-3

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Solve for x

SubEqn

LHS-x=RHS-x

3=x-3

AddEqn

LHS+3=RHS+3

6=x

RearrangeEqn

Rearrange equation

x=6

Finally, we can calculate the lengths of any of the segments by substituting the value of x into either of the given expressions. NG:& 6+3 =9 NH:& 2( 6)-3 =9 Because both of these segments have a length of 9, we know NJ does as well.