## MWSiOUX 3 years ago An isosceles trapezoid has an area of 21540 feet. What are the lengths of each base and the height?

1. pfenn1

What does the fact that it is an isosceles triangle tell you about the sides of the triangle?

2. MWSiOUX

They're the same.

3. pfenn1

Sorry, I meant isosceles trapezoid.|dw:1342964621092:dw|A trapezoid is a quadrilateral with exactly one pair of parallel sides. In a trapezoid the parallel sides are called bases. A pair of angles that share a base as a common side are called a pair of base angles. A trapezoid with the two non-parallel sides the same length is called an isosceles trapezoid. The base angles of an isosceles trapezoid are equal in measure.

4. pfenn1

The area is given by$A = \frac12(a+b)h$

5. pfenn1

Do you have enough information to solve this problem?

6. MWSiOUX

I'm not sure.

7. agentx5

I don't think you have enough information actually... The two figures below could potentially have the same area |dw:1342966291436:dw|

8. agentx5

In the same way that the can: |dw:1342966455204:dw|

9. MWSiOUX

That was all the information I was given. It's for an area of a playground.

10. agentx5

But you see what I mean with the rectangle right? What's 2m times 8m? 16m$$^2$$ right?

11. MWSiOUX

Yes. What I'm doing now is using the formula for the area of a trapezoid (.5h(b1+b2)) and made b1 = h (to simplify things a bit).

12. MWSiOUX

And then I used a graph to see what points could fit the bases more realistically. I got b1 = h = 120 and b2 = 239.

13. MWSiOUX

Would that be an isosceles trapezoid then? How could I tell?

14. MWSiOUX

$21540 = (1/2)b _{1}(b _{1}+b _{2})$ $(43080-b _{1}^{2})/b _{1}=b _{2}$

15. MWSiOUX

b1=120 and b2=239 happens to fit that.

16. agentx5

Whoa not so fast there... look: Prime factorization of your area: 21540 ft$$^2$$ = 2 * 2 * 3 * 5 * 359 * ft * ft There's going to more than one correct, whole-unit answer! That's my point, you need a bit more information. 120 + 239 = 359 1/2 * (359) * 120 = 21540, correct! But... so is: 1/2 * (139 + 220) * 120 = 21540, also correct!

17. agentx5
18. agentx5

As I said, more than one correct, whole # answer. Let roots also be include in options for answers, and you'll potentially get an unlimited set of correct answers. You need more information. If the goal was to find just one of them, then you're good. But there's more than one answer here. ;-)

19. MWSiOUX

Indeed. It's for a playground so I'd rather have a iso trap with more of a square inner area rather than a elongated, skinny inner area.

20. agentx5

If you want to find, whole number answers for these, you'll need to always write out prime factorizations as the first step. A calculus version of this problem would ask you to find the local minimum or maximum for something, or approaching a specific value (like if we needed the longest side to be no more than 200m)

21. agentx5

Make sense?

22. MWSiOUX

I was thinking about using the calculus approach but this is just for a college geometry course. So, I'm trying to think in a geometric aspect rather than a calculus aspect.

23. MWSiOUX

And yes, I understand. Thank you for your help!

24. agentx5

Ok this question got me thinking, I'm going to make a challenge question for the bored people who like mental puzzles from this original question :-D (don't worry I'll link it back @MWSiOUX )

25. MWSiOUX

Awesome!