Question

Lines AC¯¯¯¯¯¯¯¯ and BD¯¯¯¯¯¯¯¯
𝐴
𝐶
¯

𝑎
𝑛
𝑑

𝐵
𝐷
¯
are diagonals of ABCD. What value of x makes ABCD a rectangle if AC = x + 29 and BD = 4x - 1.

(1 point)
Responses

x = 15
x = 15

x = 10
x = 10

x = 6
x = 6

Answer 1

In a rectangle, the diagonals are equal in length. Therefore, we can set the lengths of the diagonals \( AC \) and \( BD \) equal to each other:

\[
x + 29 = 4x - 1
\]

To solve for \( x \), first, let's rearrange the equation:

1. Subtract \( x \) from both sides:
\[
29 = 4x - x - 1
\]
Which simplifies to:
\[
29 = 3x - 1
\]

2. Next, add \( 1 \) to both sides:
\[
29 + 1 = 3x
\]
Thus:
\[
30 = 3x
\]

3. Finally, divide by \( 3 \):
\[
x = 10
\]

So, the value of \( x \) that makes \( ABCD \) a rectangle is **\( x = 10 \)**.