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In this diagram, NPQR is a rectangle. What is the length, in units, of NQ?​

In this diagram, NPQR is a rectangle. What is the length, in units, of NQ?​-example-1

1 Answer

6 votes

Answer: 14 units

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Step-by-step explanation:

Rule: The diagonals of any rectangle are the same length.

Because of that rule, we can say

NQ = PR

(NK) + (KQ) = (PK) + (PR)

(x+4) + (2y+5) = (2x+1) + (4y+3)

x+2y+9 = 2x+4y+4

x+2y-2x-4y = 4-9

-x-2y = -5

x+2y = 5

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Rule: The diagonals of any rectangle cut each other in half (aka bisect each other)

So the pieces NK and KQ are the same length.

NK = KQ

x+4 = 2y+5

x = 2y+5-4

x = 2y+1

Plug this into the previous equation we found. Solve for y.

x+2y = 5

(2y+1) + 2y = 5

4y+1 = 5

4y = 5-1

4y = 4

y = 4/4

y = 1

Then use this to find x

x = 2y+1

x = 2*1+1

x = 3

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We now have enough info to find the length of each segment.

  • NK = x+4 = 3+4 = 7
  • KQ = 2y+5 = 2*1+5 = 7
  • NQ = NK+KQ = 7+7 = 14 which is the final answer

Furthermore,

  • PK = 2x+1 = 2*3+1 = 7
  • KR = 4y+3 = 4*1+3 = 7
  • PR = PK+KR = 7+7 = 14

We get identical segment lengths, which helps confirm the correct answer.

User Pawan Choudhary
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