Question

which of the following statements are true? select all that apply.
a) the perimeter of triangle 1 is x + 22.
b) the perimeter of triangle 1 is -x + 38.
c) the perimeter of triangle 2 is 13x - 20.
d) the perimeter of triangle 2 is 8x - 6.
e) if x = 4, triangle 1 has the greater perimeter.
f) if x = 5, triangle 2 has the greater perimeter.

Explanation:

Step1: Calculate the perimeter of Triangle 1

The perimeter \(P_1\) of Triangle 1 is the sum of its side lengths.

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Step2: Calculate the perimeter of Triangle 2

The perimeter \(P_2\) of Triangle 2 is the sum of its side lengths.

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Step3: Check option A

If \(P_1=x + 22\), then \(-x + 38=x + 22\). Solving for \(x\):

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Since \(P_1=-x + 38\) (not \(x + 22\)), option A is false.

Step4: Check option B

Since \(P_1=-x + 38\), option B is true.

Step5: Check option C

Since \(P_2=13x-20\), option C is true.

Step6: Check option D

If \(P_2=8x - 6\), then \(13x-20=8x - 6\). Solving for \(x\):

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Since \(P_2=13x-20\) (not \(8x - 6\)), option D is false.

Step7: Check option E

When \(x = 4\)
\(P_1=-4 + 38=34\)
\(P_2=13\times4-20=52 - 20 = 32\)
Since \(P_1>P_2\), option E is false.

Step8: Check option F

When \(x = 5\)
\(P_1=-5 + 38=33\)
\(P_2=13\times5-20=65 - 20=45\)
Since \(P_2>P_1\), option F is true.

Answer:

B. The perimeter of Triangle 1 is \(-x + 38\), C. The perimeter of Triangle 2 is \(13x - 20\), F. If \(x = 5\), Triangle 2 has the greater perimeter.