Question

In parallelogram ABCD , BE=7x−2 and DE=x^2−10.Segment AC intersects segment BD at point E. What is BD ? Enter your answer in the box.

Answer 1

Length of BD = 108 units

Explanation:

Given: In parallelogram ABCD , BE = 7x-2 and DE =
`x^2-10`.

Segment AC intersects segment BD at point E.

Properties of parallelograms:-

• Opposite sides are congruent (AB=CD)
• Opposite angels are congruent (D=B).
• Also, consecutive angles are supplementary (A + D = 180°).

• Diagonals of a parallelogram bisect each other.

• Every diagonal of a parallelogram separates it into two congruent.

The diagonal BD;

BD = BE + ED

since, by properties of parallelogram;

BE = ED ......[1]

Substitute the given values of BE and DE in [1] to solve for x;

`7x-2=x^2-10`

or

`x^2-10-7x+2=0`

`x^2-7x-8=0`

`x^2-8x+x-8=0`

`x(x-8)+1(x-8)=0`

(x-8)(x+1) = 0

equate each factor equals to 0 we get;

x = 8 and x = -1

Since sides are always in positive,

⇒x =8

Then;

BE = 7x -2 = 7(8) -2 = 56-2 = 54 units.

To find the Length of BD:

As we know;

BD = BE + ED = BE + BE = 2BE [Since, BE = ED]

BD =
`2 * 54 =108` units

Therefore, the length of BD is 108 units.