Question

1. which formulas can be used to find the perimeter ( p ) of a rectangle with length ( ell ) and width ( w )? select all that apply.

a ( p = ell + w )
b ( p = ell w )
c ( p = 2ell + 2w )
d ( p = (2ell)(2w) )
e ( p = 2(ell + w) )
f ( p = 4ell w )

Explanation:

Step1: Recall Rectangle Perimeter Definition

A rectangle has two lengths and two widths. The perimeter is the sum of all sides. So, \( P = \ell + \ell + w + w \).

Step2: Simplify the Sum

Combine like terms: \( \ell + \ell + w + w = 2\ell + 2w \).

Step3: Factor the Expression

Factor out the 2 from \( 2\ell + 2w \): \( 2(\ell + w) \).

Step4: Analyze Each Option

• Option A: \( P = \ell + w \) is the sum of one length and one width, not the perimeter. Incorrect.
• Option B: \( P = \ell w \) is the area formula (length × width), not perimeter. Incorrect.
• Option C: \( P = 2\ell + 2w \) matches the simplified sum of all sides. Correct.
• Option D: \( P = (2\ell)(2w) = 4\ell w \), which is not related to perimeter. Incorrect.
• Option E: \( P = 2(\ell + w) \) matches the factored form. Correct.
• Option F: \( P = 4\ell w \) is incorrect (not a perimeter formula).

Answer:

C. \( P = 2\ell + 2w \), E. \( P = 2(\ell + w) \)