Question

1. in δdef, de = 7x + 1, ef = 11x - 18, and df = 5x + 13. if the perimeter of δdef is 88, which of the following lists the angles in order from least to greatest?

a. ∠d, ∠e, ∠f
b. ∠d, ∠f, ∠e
c. ∠f, ∠d, ∠e
d. ∠e, ∠f, ∠d

Explanation:

Step1: Find the value of x

The perimeter of a triangle is the sum of its three sides. So, \( DE + EF + DF = 88 \). Substituting the given expressions: \( (7x + 1) + (11x - 18) + (5x + 13) = 88 \). Combine like terms: \( 7x + 11x + 5x + 1 - 18 + 13 = 88 \), which simplifies to \( 23x - 4 = 88 \). Add 4 to both sides: \( 23x = 92 \). Divide by 23: \( x = 4 \).

Step2: Calculate the lengths of the sides

• \( DE = 7x + 1 = 7(4) + 1 = 29 \)
• \( EF = 11x - 18 = 11(4) - 18 = 26 \)
• \( DF = 5x + 13 = 5(4) + 13 = 33 \)

Step3: Relate side lengths to angles

In a triangle, the larger the side, the larger the angle opposite it. The sides (from shortest to longest) are \( EF = 26 \), \( DE = 29 \), \( DF = 33 \). The angles opposite these sides are \( \angle D \) (opposite \( EF \)), \( \angle F \) (opposite \( DE \)), and \( \angle E \) (opposite \( DF \)). So the angles from least to greatest are \( \angle D \), \( \angle F \), \( \angle E \).

Answer:

B. \( \angle D \), \( \angle F \), \( \angle E \)