Question

1. d(-2, 3), e(5, 5), f(-4, 10)

directions: solve each problem given the information.

1. if \\( \triangle jkl \\) is an equilateral triangle, \\( jk = 13x + 5 \\), \\( kl = 17x - 19 \\), and \\( jl = 8x + 35 \\), find the value of \\( x \\) and the measure of each side.
2. \\( \triangle qrs \\) is an equilateral triangle. if \\( qr \\) is seven less than twice \\( x \\), \\( rs \\) is 61 less than five times \\( x \\), and \\( qs \\) is 11 more than \\( x \\), find the value of \\( x \\) and the measure of each side.

Explanation:

Problem 12:

Step1: Set sides equal (equilateral triangle)

In an equilateral triangle, all sides are equal. So, \( JK = KL \).
Given \( JK = 13x + 5 \) and \( KL = 17x - 19 \), we set them equal:
\( 13x + 5 = 17x - 19 \)

Step2: Solve for \( x \)

Subtract \( 13x \) from both sides:
\( 5 = 4x - 19 \)
Add 19 to both sides:
\( 24 = 4x \)
Divide by 4:
\( x = 6 \)

Step3: Find side length

Substitute \( x = 6 \) into \( JK \):
\( JK = 13(6) + 5 = 78 + 5 = 83 \)
Check with \( KL \):
\( KL = 17(6) - 19 = 102 - 19 = 83 \)
Check with \( JL \):
\( JL = 8(6) + 35 = 48 + 35 = 83 \)

Problem 13:

Step1: Define sides (equilateral triangle)

Let’s define the sides:

• \( QR = 2x - 7 \) (seven less than twice \( x \))
• \( RS = 5x - 61 \) (61 less than five times \( x \))
• \( QS = x + 11 \) (11 more than \( x \))

In an equilateral triangle, \( QR = RS \).

Step2: Solve for \( x \)

Set \( QR = RS \):
\( 2x - 7 = 5x - 61 \)
Subtract \( 2x \) from both sides:
\( -7 = 3x - 61 \)
Add 61 to both sides:
\( 54 = 3x \)
Divide by 3:
\( x = 18 \)

Step3: Find side length

Substitute \( x = 18 \) into \( QR \):
\( QR = 2(18) - 7 = 36 - 7 = 29 \)
Check \( RS \):
\( RS = 5(18) - 61 = 90 - 61 = 29 \)
Check \( QS \):
\( QS = 18 + 11 = 29 \)

Problem 11 (Distance between points):

Step1: Recall distance formula

The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step2: Calculate \( DE \), \( EF \), \( FD \) (if needed, but since it's not specified, let's do one as example, say \( DE \))

For \( D(-2, 3) \) and \( E(5, 5) \):
\( x_1 = -2, y_1 = 3 \); \( x_2 = 5, y_2 = 5 \)
\( d = \sqrt{(5 - (-2))^2 + (5 - 3)^2} = \sqrt{(7)^2 + (2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.28 \)

Answer:

s:

• Problem 12: \( x = 6 \), each side = \( 83 \)
• Problem 13: \( x = 18 \), each side = \( 29 \)
• Problem 11 (example \( DE \)): \( \sqrt{53} \) (or approximate \( 7.28 \))