Question

Abce is an isosceles trapezoid and acde is a rectangle. ab = 10 and ec = 20. what is the length of ae?

Answer 1

Draw the isosceles trapezoid ABCE, where the length of the sides AB and EC are respectively 10 and 20.

It is given that ACDE is a rectangle (draw the diagonal AC, from C draw a line parallel to the side AE of the trapezoid, from E draw another line parallel to AC, and label the point of intersection with D).

Look at the right triangle EAC. Let the length of EA be x, and the length of AC be y. So we have,

x^2 + y^2 = 20^2 or
y^2 = 20^2 - x^2.

From A, draw the perpendicular AF to EC. Let the length of AF be z.
Look at the right triangle EFA, where the length of EF is 5. Here we have,

x^2 = z^2 + 5^2 or
z^2 = x^2 - 5^2

Look at the right triangle CFA, where the length of CF is 15. Here we have,

y^2 = z^2 + 15^2 or
z^2 = y^2 - 15^2

So we have:

z^2 = y^2 - 15^2
y^2 = 20^2 - x^2

z^2 = y^2- 15^2 or
z^2 = 20^2 - x^2 - 15^2

Also we have:

z^2 = x^2 - 5^2
z^2 = 20^2 - x^2 - 15^2

Then,

x^2 - 5^2 = 20^2 - x^2 - 15^2
2x^2 = 20^2 - 15^2 + 5^2
2x^2 = 400 - 225 + 25
2x^2 = 200
x^2 = 100
x = √100 (ignore the negative sign, because the length is positive)
x = 10

Thus, the length of AE is 10.

Answer 2

Draw the isosceles trapezoid ABCE, where the length of the sides AB and EC are respectively $10$ and $20$.

It is given that ACDE is a rectangle (draw the diagonal AC, from C draw a line parallel to the side AE of the trapezoid, from E draw another line parallel to AC, and label the point of intersection with D).

Look at the right triangle EAC. Let the length of EA be x, and the length of AC be y. So we have,

$x^2 + y^2 = 20^2$

$y^2 = 20^2 - x^2.$

From A, draw the perpendicular AF to EC. Let the length of AF be z.

Look at the right triangle EFA, where the length of EF is $5$. Here we have,

$x^2 = z^2 + 5^2$ or

$z^2 = x^2 - 5^2$

Look at the right triangle CFA, where the length of CF is $15$. Here we have,

$y^2 = z^2 + 15^2$ or

$z^2 = y^2 - 15^2$

So we have:

$z^2 = y^2 - 15^2$

$y^2 = 20^2 - x^2$

$z^2 = y^2- 15^2$ or

$z^2 = 20^2 - x^2 - 15^2$

Also, we have:

$z^2 = x^2 - 5^2$

$z^2 = 20^2 - x^2 - 15^2$

Then,

$x^2 - 5^2 = 20^2 - x^2 - 15^2$

$2x^2 = 20^2 - 15^2 + 5^2$

$2x^2 = 400 - 225 + 25$

$2x^2 = 200$

$x^2 = 100$

$x = \sqrt{100}$ (ignore the negative sign, because the length is positive)

$x = 10$

Thus, the length of AE is $10$.

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