Question

pqrs is a rectangle. find the value of x and the length of each diagonal.
pr = 10x - 4 and sq = 7x + 5
x =
pr = sq =

Explanation:

Step1: Recall the property of a rectangle's diagonals

In a rectangle, the diagonals are equal in length. So, \( PR = SQ \).

Step2: Set up the equation

Given \( PR = 10x - 4 \) and \( SQ = 7x + 5 \), we set them equal: \( 10x - 4 = 7x + 5 \).

Step3: Solve for \( x \)

Subtract \( 7x \) from both sides: \( 10x - 7x - 4 = 7x - 7x + 5 \), which simplifies to \( 3x - 4 = 5 \). Then add 4 to both sides: \( 3x - 4 + 4 = 5 + 4 \), so \( 3x = 9 \). Divide both sides by 3: \( x = \frac{9}{3} = 3 \).

Step4: Find the length of the diagonals

Substitute \( x = 3 \) into \( PR = 10x - 4 \): \( PR = 10(3) - 4 = 30 - 4 = 26 \). Since \( PR = SQ \), \( SQ = 26 \) as well.

Answer:

\( x = 3 \), \( PR = SQ = 26 \)