Exercise 34 Page 523

Let's analyze the given quadrilateral to find the length JQ. Keep in mind we have been told that the figure is a rectangle.

Since the given figure is a rectangle, we know that all the segments of the diagonals are congruent. JQ=QM = NQ=QK We can use this fact to substitute the given expressions, NQ=2x+3 and QK=5x-9, to find the value of x. NQ=QK ⇔ 2x+3 = 5x-9 We can use this equation to solve for x.

2x+3=5x-9

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Solve for x

SubEqn

LHS-5x=RHS-5x

- 3x +3=- 9

SubEqn

LHS-3=RHS-3

- 3x=- 12

DivEqn

.LHS /- 3.=.RHS /- 3.

x=4

Knowing the value of x, we can find the length of one of the given segments. Let's find NQ.

NQ = 2x+3

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Substitute values and evaluate

Substitute

x= 4

NQ = 2 * 4+3

Multiply

Multiply

NQ = 8+3

AddTerms

Add terms

NQ = 11

Since all the segments are equal, JQ is 11.