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The perimeters of the triangles shown are equal. Find the side lengths of each triangle.

A
B
C
2x+6
2x+5
2x
2x
2x
3x+4
P
Q
R
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Part 1
AB overbarequals
  
enter your response here
PQ overbarequals
  
enter your response here
BC overbarequals
  
enter your response here
QR overbarequals
  
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AC overbarequals
  
enter your response here
PR overbarequals
  
enter your response here
​(Type a whole​ number.)

1 answer

To find the side lengths of each triangle given that the perimeters are equal, we need to set up an equation based on the perimeters of triangles A and B.

Triangle A:

  • Side lengths: \(2x + 6\), \(2x + 5\), and \(2x\).

The perimeter of triangle A is: \[ P_A = (2x + 6) + (2x + 5) + (2x) = 6x + 11 \]

Triangle B:

  • Side lengths: \(2x\), \(3x + 4\), and \(2x\).

The perimeter of triangle B is: \[ P_B = (2x) + (3x + 4) + (2x) = 7x + 4 \]

Setting the perimeters equal:

Since the perimeters are equal, we set \(P_A\) equal to \(P_B\): \[ 6x + 11 = 7x + 4 \]

Solving for \(x\):

  1. Rearrangement: \[ 11 - 4 = 7x - 6x \] \[ 7 = x \]

Finding side lengths:

Now, substitute \(x = 7\) back into the expressions for the side lengths.

Triangle A:

  • \(AB = 2x + 6 = 2(7) + 6 = 14 + 6 = 20\)
  • \(BC = 2x + 5 = 2(7) + 5 = 14 + 5 = 19\)
  • \(AC = 2x = 2(7) = 14\)

Triangle B:

  • \(PQ = 2x = 2(7) = 14\)
  • \(QR = 3x + 4 = 3(7) + 4 = 21 + 4 = 25\)
  • \(PR = 2x = 2(7) = 14\)

Summary of Side Lengths:

  • Triangle A:

    • \(AB = 20\)
    • \(BC = 19\)
    • \(AC = 14\)

  • Triangle B:

    • \(PQ = 14\)
    • \(QR = 25\)
    • \(PR = 14\)

Thus, the side lengths of each triangle are:

  • For Triangle A:

    • \(AB = 20\)
    • \(BC = 19\)
    • \(AC = 14\)

  • For Triangle B:

    • \(PQ = 14\)
    • \(QR = 25\)
    • \(PR = 14\)