Exercise 50 Page 343

Practice makes perfect

We are given the following diagram and asked to find the measure of TQ.

As we can see, 2x-6 represents the length of TQ. Therefore, to find TQ, we need first to find the value of x.

Finding x

From the diagram, we see that SQ intersects RT at the midpoint S. Also, it is perpendicular to RT. Thereby, SQ is the perpendicular bisector of RT. Let's use the Perpendicular Bisector Theorem.

Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

By this theorem, Q is equidistant from the endpoints of RT. This means that RQ and TQ have the same lengths. From the diagram, we know that the length of RQ is x+3 and the length of TQ is 2x-6. Let's set these expressions equal. x+3= 2x-6 Now, we can solve this equation for x.

x+3=2x-6

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Solve for x

SubEqn

LHS-x=RHS-x

3=x-6

AddEqn

LHS+6=RHS+6

9=x

RearrangeEqn

Rearrange equation

x=9