Question

7 multiple answer 4 points
the side lengths of two triangles are shown.
which of the following statements are true? select all that apply.
triangle 1
$3x$
$6$
$2(-2x + 16)$
triangle 2
$3x$
$8$
$2(5x - 14)$
the perimeter of triangle 2 is $13x$-$20$.
the perimeter of triangle 1 is $-x$+$38$.
if $x = 4$,triangle 1 has the greater perimeter.
the perimeter of triangle 1 is $x$+$22$.
the perimeter of triangle 2 is $8x$-$6$.
if $x = 5$, triangle 2 has the greater perimeter.

Explanation:

Step1: Calculate Triangle 1 perimeter

Sum its side lengths:

$$\begin{align*} P_1&=3x + 6 + 2(-2x+16)\\ &=3x + 6 -4x +32\\ &=-x + 38 \end{align*}$$

Step2: Calculate Triangle 2 perimeter

Sum its side lengths:

$$\begin{align*} P_2&=3x + 8 + 2(5x-14)\\ &=3x + 8 +10x -28\\ &=13x -20 \end{align*}$$

Step3: Test x=4 for perimeters

Substitute $x=4$:
$P_1= -4 +38=34$, $P_2=13(4)-20=32$
$34>32$, so Triangle 1 is larger.

Step4: Test x=5 for perimeters

Substitute $x=5$:
$P_1= -5 +38=33$, $P_2=13(5)-20=45$
$45>33$, so Triangle 2 is larger.

Answer:

1. The perimeter of Triangle 2 is 13x-20.
2. The perimeter of Triangle 1 is -x+38.
3. If x = 4, Triangle 1 has the greater perimeter.
4. If x = 5, Triangle 2 has the greater perimeter.