Question

abcd is a rectangle. if de = 4x + 1, eb = 12x - 31, and cd = 28, find bc. round your answer to the nearest tenths place (one decimal place).

Explanation:

Step1: Set DE = EB (diagonals bisect each other in rectangle)

In a rectangle, the diagonals bisect each other, so \( DE = EB \).
Given \( DE = 4x + 1 \) and \( EB = 12x - 31 \), we set up the equation:
\( 4x + 1 = 12x - 31 \)

Step2: Solve for x

Subtract \( 4x \) from both sides:
\( 1 = 8x - 31 \)
Add 31 to both sides:
\( 32 = 8x \)
Divide both sides by 8:
\( x = 4 \)

Step3: Find length of diagonal DB

First, find \( DE \) by substituting \( x = 4 \) into \( DE = 4x + 1 \):
\( DE = 4(4) + 1 = 17 \)
Since \( DE = EB \), \( EB = 17 \), so \( DB = DE + EB = 17 + 17 = 34 \)

Step4: Use Pythagorean theorem to find BC

In rectangle \( ABCD \), triangle \( BCD \) is a right triangle with \( CD = 28 \) (leg), \( DB = 34 \) (hypotenuse), and \( BC \) (leg). By Pythagorean theorem \( BC = \sqrt{DB^2 - CD^2} \)
\( BC = \sqrt{34^2 - 28^2} = \sqrt{1156 - 784} = \sqrt{372} \approx 19.3 \)

Answer:

\( 19.3 \)