Question

1. the lengths (in meters) of two sides of a regular triangle are represented by the expressions 3x - 5 and x + 9. find the length of a side of the triangle.
2. the expressions 5x + 13 and 10x - 7 represent the lengths (in inches) of two sides of an equilateral octagon. find the length of a side of the octagon.
3. the expressions 7x + 34 and 11x - 14 represent the lengths (in feet) of two sides of a regular hexagon. find the length of a side of the hexagon.

Explanation:

Problem 8:

Step1: Set expressions equal (regular triangle)

In a regular triangle, all sides are equal. So, set \( 3x - 5 = x + 9 \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 3x - x - 5 = 9 \) → \( 2x - 5 = 9 \).
Add 5 to both sides: \( 2x = 9 + 5 \) → \( 2x = 14 \).
Divide by 2: \( x = \frac{14}{2} = 7 \).

Step3: Find side length

Substitute \( x = 7 \) into \( 3x - 5 \): \( 3(7) - 5 = 21 - 5 = 16 \).
(Or into \( x + 9 \): \( 7 + 9 = 16 \), same result.)

Step1: Set expressions equal (equilateral octagon)

In an equilateral octagon, all sides are equal. So, set \( 5x + 13 = 10x - 7 \).

Step2: Solve for \( x \)

Subtract \( 5x \) from both sides: \( 13 = 5x - 7 \).
Add 7 to both sides: \( 13 + 7 = 5x \) → \( 20 = 5x \).
Divide by 5: \( x = \frac{20}{5} = 4 \).

Step3: Find side length

Substitute \( x = 4 \) into \( 5x + 13 \): \( 5(4) + 13 = 20 + 13 = 33 \).
(Or into \( 10x - 7 \): \( 10(4) - 7 = 40 - 7 = 33 \), same result.)

Step1: Set expressions equal (regular hexagon)

In a regular hexagon, all sides are equal. So, set \( 7x + 34 = 11x - 14 \).

Step2: Solve for \( x \)

Subtract \( 7x \) from both sides: \( 34 = 4x - 14 \).
Add 14 to both sides: \( 34 + 14 = 4x \) → \( 48 = 4x \).
Divide by 4: \( x = \frac{48}{4} = 12 \).

Step3: Find side length

Substitute \( x = 12 \) into \( 7x + 34 \): \( 7(12) + 34 = 84 + 34 = 118 \).
(Or into \( 11x - 14 \): \( 11(12) - 14 = 132 - 14 = 118 \), same result.)

Answer:

The length of a side of the triangle is \( \boldsymbol{16} \) meters.

Problem 9: