Given the rectangle [tex]\( ABCD \)[/tex] with diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex]:
- Length of diagonal [tex]\( AC = 8 + 6x \)[/tex]
- Length of diagonal [tex]\( BD = 4 + 7x \)[/tex]
By definition, in a rectangle, the diagonals are equal in length. Therefore, [tex]\( AC = BD \)[/tex].
Let's set up the equation based on the equality of the diagonals:
[tex]\[ 8 + 6x = 4 + 7x \][/tex]
To find [tex]\( x \)[/tex], let's solve this equation step-by-step:
1. Subtract [tex]\( 4 \)[/tex] from both sides:
[tex]\[ 8 + 6x - 4 = 7x \][/tex]
2. Simplify the left-hand side:
[tex]\[ 4 + 6x = 7x \][/tex]
3. Subtract [tex]\( 6x \)[/tex] from both sides to isolate x:
[tex]\[ 4 = x \][/tex]
So, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = 4 \][/tex]
Now, let's compute the lengths of diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] using [tex]\( x = 4 \)[/tex].
For [tex]\( AC \)[/tex]:
[tex]\[ AC = 8 + 6x = 8 + 6(4) = 8 + 24 = 32 \][/tex]
For [tex]\( BD \)[/tex]:
[tex]\[ BD = 4 + 7x = 4 + 7(4) = 4 + 28 = 32 \][/tex]
Therefore:
[tex]\[ x = 4 \][/tex]
[tex]\[ AC = 32 \][/tex]
[tex]\[ BD = 32 \][/tex]
So, the segments that have equal length are:
[tex]\[ AC = BD \][/tex]
In summary:
Given rectangle [tex]\( ABCD \)[/tex],
[tex]\[ AC = 8 + 6x \][/tex]
[tex]\[ BD = 4 + 7x \][/tex]
Using the definition of a rectangle, the segments that have equal length are [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex].
Solving the equation [tex]\( 8 + 6x = 4 + 7x \)[/tex], we get:
[tex]\[ x = 4 \][/tex]
[tex]\[ AC = 32 \][/tex]
[tex]\[ BD = 32 \][/tex]
