Question

in the diagram below, figure abcd is a rectangle. use the figure for questions 1–6.

1. what is the value of x?
2. what is the value of y?
3. what is the measure of \\(\overline{ae}\\)?
4. what is the measure of \\(\overline{eb}\\)?
5. what is the measure of \\(\overline{ac}\\)?
6. what is the measure of \\(\overline{db}\\)?

Explanation:

Step1: Set equal segments (x)

In a rectangle, diagonals bisect each other, so $3x+3=4x$.

Step2: Solve for x

Subtract $3x$ from both sides:
$3x+3-3x=4x-3x$
$x=3$

Step3: Set equal segments (y)

Diagonals bisect each other, so $3y+1.5=4y-2$.

Step4: Solve for y

Subtract $3y$ and add 2 to both sides:
$3y+1.5-3y+2=4y-2-3y+2$
$y=3.5$

Step5: Calculate $\overline{AE}$

Substitute $y=3.5$ into $3y+1.5$:
$3(3.5)+1.5=10.5+1.5=12$

Step6: Calculate $\overline{EB}$

Substitute $x=3$ into $4x$:
$4(3)=12$

Step7: Calculate $\overline{AC}$

Diagonal $AC=2\times AE=2\times12=24$

Step8: Calculate $\overline{DB}$

In a rectangle, diagonals are equal, so $DB=AC=24$

Answer:

1. $x=3$
2. $y=3.5$
3. $\overline{AE}=12$
4. $\overline{EB}=12$
5. $\overline{AC}=24$
6. $\overline{DB}=24$