Question

each figure is a regular polygon. expressions are given for two side lengths. find the value of x.
19.
20.
21.
22.
23.
24.

Explanation:

Problem 19 (Regular Pentagon)

Step1: Set sides equal (regular polygon)

In a regular pentagon, all sides are equal. So, \( 3x - 1 = x + 5 \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 2x - 1 = 5 \).
Add 1 to both sides: \( 2x = 6 \).
Divide by 2: \( x = 3 \).

Step1: Set sides equal (regular polygon)

In an equilateral triangle, all sides are equal. So, \( 5x - 9 = 2x + 15 \).

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides: \( 3x - 9 = 15 \).
Add 9 to both sides: \( 3x = 24 \).
Divide by 3: \( x = 8 \).

Step1: Set sides equal (regular polygon)

In a regular quadrilateral (rhombus), all sides are equal. So, \( 7x + 8 = 14x - 34 \).

Step2: Solve for \( x \)

Subtract \( 7x \) from both sides: \( 8 = 7x - 34 \).
Add 34 to both sides: \( 42 = 7x \).
Divide by 7: \( x = 6 \).

Answer:

\( x = 3 \)

Problem 20 (Regular Triangle/Equilateral Triangle)