Question

a polygon is regular if each of its sides has the same length. find the... the regular polygon. 18. triangle with sides ( 5 - 2x ) and ( -4x + 9 ) 19. pentagon with sides ( 3(x - 1) ) and ( 5x - 6 ) writing write a linear equation that has (a) no solution and...

Explanation:

Problem 18 (Regular Triangle)

Step1: Set side lengths equal

In a regular triangle, all sides are equal. So, \( 5 - 2x = -4x + 9 \).

Step2: Solve for \( x \)

Add \( 4x \) to both sides: \( 5 - 2x + 4x = 9 \) → \( 5 + 2x = 9 \).
Subtract 5: \( 2x = 9 - 5 \) → \( 2x = 4 \).
Divide by 2: \( x = \frac{4}{2} = 2 \).

Step3: Find side length

Substitute \( x = 2 \) into \( 5 - 2x \): \( 5 - 2(2) = 5 - 4 = 1 \).
(Or into \( -4x + 9 \): \( -4(2) + 9 = -8 + 9 = 1 \).)

Problem 19 (Regular Pentagon)

Step1: Set side lengths equal

In a regular pentagon, all sides are equal. So, \( 3(x - 1) = 5x - 6 \).

Step2: Solve for \( x \)

Expand left side: \( 3x - 3 = 5x - 6 \).
Subtract \( 3x \): \( -3 = 2x - 6 \).
Add 6: \( 3 = 2x \).
Divide by 2: \( x = \frac{3}{2} = 1.5 \).

Step3: Find side length

Substitute \( x = 1.5 \) into \( 3(x - 1) \): \( 3(1.5 - 1) = 3(0.5) = 1.5 \).
(Or into \( 5x - 6 \): \( 5(1.5) - 6 = 7.5 - 6 = 1.5 \).)

Answer:

s:

1. \( x = 2 \), side length = \( 1 \)
2. \( x = 1.5 \), side length = \( 1.5 \)