Question

in parallelogram ( abcd ), diagonals ( overline{ac} ) and ( overline{bd} ) intersect at point ( e ), ( be = x^{2}-21 ), and ( de = 4x ). what is ( bd )? enter your answer in the box. ( bd=square ) units

Explanation:

Step1: Use the property of parallelogram diagonals

In a parallelogram, the diagonals bisect each other. So \(BE = DE\).
$$x^{2}-21 = 4x$$

Step2: Rearrange the equation to standard quadratic form

$$x^{2}-4x - 21=0$$

Step3: Factor the quadratic equation

$$(x - 7)(x+ 3)=0$$

Step4: Solve for \(x\)

Set each factor equal to zero:

• \(x - 7=0\Rightarrow x = 7\)
• \(x+3=0\Rightarrow x=-3\)

Since length cannot be negative, \(x = 7\).

Step5: Find \(BE\) and \(DE\)

• \(BE=x^{2}-21=(7)^{2}-21=49 - 21=28\)
• \(DE = 4x=4\times7 = 28\)

Step6: Calculate \(BD\)

Since \(BD=BE + DE\), \(BD=28+28\)

Answer:

\(56\)