Question

In parallelogram ABCD , diagonals AC¯¯¯¯¯ and BD¯¯¯¯¯ intersect at point E, AE=2x2−3x , and CE=x2+4 .

What is AC ?




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units

Answer 1

The length of AC is either 10 units or 40 units.

Explanation:

it is given that parallelogram ABCD and diagonals AC and BD intersect at point E.

According to the property of parallelogram, the diagonals are intersecting each other at their midpoint.

`AE=CE`

`2x^2-3x=x^2+4`

`2x^2-3x-x^2-4=0`

`x^2-3x-4=0`

`x^2-4x+x-4=0`

`x(x-4)+1(x-4)=0`

`(x-4)(x+1)=0`

By zero product property, equate each factor equal to 0. So the value of x is 4 and -1.

`AC=2CE`

`AC=2(x^2+4)`

Let he value of x=4.

`AC=2(4^2+4)`

`AC=2(20)`

`AC=40`

Therefore the length of AC is 40.

Let he value of x=-1.

`AC=2((-1)^2+4)`

`AC=2(5)`

`AC=10`

Therefore the length of AC is 10.

Answer 2

If the diagonals of the parallelogram intersect with each other, this means that the diagonals bisect each other. Thus,

AE = CE

Substituting the equations,

2x² - 3x = x² + 4

The values of x from the equation are equal to 4 and -1.

AE = 2(4)² - 3(4) = 20

Hence, AC is equal to 40.

Answer: 40 units