Question

1. the perimeter of a rectangle is 32 inches.

• the rectangle has a width of 4 inches.
• the rectangle has a length of 2.5x inches.

which equation can be used to find the value of x?
a. 4 + 2.5x = 32
b. 8 + 5x = 32
c. 4(2.5x) = 32
d. 8(5x) = 32

1. lakeside park is redesigning its layout by adding new pathways. three of these paths will intersect, forming a triangle, as shown in the diagram.

what are the measures of ∠1 and ∠2?
a. m∠1 = 40°, m∠2 = 30°
b. m∠1 = 150°, m∠2 = 40°
c. m∠1 = 110°, m∠2 = 30°
d. m∠1 = 30°, m∠2 = 50°

Explanation:

Step1: Recall the perimeter formula of a rectangle

The perimeter formula of a rectangle is \(P = 2(l + w)\), where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width. Given \(P=32\) inches, \(w = 4\) inches, and \(l=2.5x\) inches.
Substitute into the formula: \(32=2(4 + 2.5x)\)

Step2: Simplify the equation

First, distribute the \(2\) on the right - hand side: \(32=8 + 5x\)

for the second question:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
For \(\angle2\): Since an exterior angle of the triangle is \(150^{\circ}\), and one non - adjacent interior angle is \(120^{\circ}\) (not given in the problem, but using the fact that \(\angle3\) and \(40^{\circ}\) form a linear pair, \(\angle3 = 140^{\circ}\) is wrong, actually, for \(\angle2\), using the exterior - angle property: \(\angle2=150^{\circ}- 120^{\circ}=30^{\circ}\) (wait, no, correct way: \(\angle2\) and \(150^{\circ}\) form a linear pair, so \(\angle2 = 180^{\circ}-150^{\circ}=30^{\circ}\). Then, using the sum of angles in a triangle (\(180^{\circ}\)), if one angle is \(40^{\circ}\) and \(\angle2 = 30^{\circ}\), then \(\angle1=180^{\circ}-(40^{\circ}+ 30^{\circ})=110^{\circ}\)

Answer:

B. \(8 + 5x=32\)