Question

1. if $\triangle pqr$ is an equilateral triangle, $pq = 18x + 1$, $qr = 24x - 17$, and $pr = 15x + 10$, find $x$ and the measure of each side.
2. $\triangle stu$ is an equilateral triangle. if $st$ is one less than twice $x$, $su$ is 37 less than five times $x$, and $tu$ is 11 more than $x$, find $x$ and the measure of each side.

Explanation:

Problem 6:

Step1: Recall equilateral triangle property

In an equilateral triangle, all sides are equal. So, \( PQ = QR = PR \). Let's set \( PQ = QR \):
\( 18x + 1 = 24x - 17 \)

Step2: Solve for \( x \)

Subtract \( 18x \) from both sides:
\( 1 = 6x - 17 \)

Add 17 to both sides:
\( 18 = 6x \)

Divide by 6:
\( x = 3 \)

Step3: Find side length

Substitute \( x = 3 \) into \( PQ \):
\( PQ = 18(3) + 1 = 54 + 1 = 55 \)

Check with \( QR \):
\( QR = 24(3) - 17 = 72 - 17 = 55 \)

Check with \( PR \):
\( PR = 15(3) + 10 = 45 + 10 = 55 \)

Problem 7:

Step1: Recall equilateral triangle property

In \( \triangle STU \), \( ST = SU = TU \).

Define sides:
\( ST = 2x - 1 \)
\( SU = 5x - 37 \)
\( TU = x + 11 \)

Set \( ST = SU \):
\( 2x - 1 = 5x - 37 \)

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides:
\( -1 = 3x - 37 \)

Add 37 to both sides:
\( 36 = 3x \)

Divide by 3:
\( x = 12 \)

Step3: Find side length

Substitute \( x = 12 \) into \( ST \):
\( ST = 2(12) - 1 = 24 - 1 = 23 \)

Check \( SU \):
\( SU = 5(12) - 37 = 60 - 37 = 23 \)

Check \( TU \):
\( TU = 12 + 11 = 23 \)

Answer:

(Problem 6):
\( x = 3 \), each side is \( 55 \)