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MWSiOUX
An isosceles trapezoid has an area of 21540 feet. What are the lengths of each base and the height?
What does the fact that it is an isosceles triangle tell you about the sides of the triangle?
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.
The area is given by\[A = \frac12(a+b)h\]
Do you have enough information to solve this problem?
I don't think you have enough information actually... The two figures below could potentially have the same area |dw:1342966291436:dw|
In the same way that the can: |dw:1342966455204:dw|
That was all the information I was given. It's for an area of a playground.
But you see what I mean with the rectangle right? What's 2m times 8m? 16m\(^2\) right?
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).
And then I used a graph to see what points could fit the bases more realistically. I got b1 = h = 120 and b2 = 239.
Would that be an isosceles trapezoid then? How could I tell?
\[21540 = (1/2)b _{1}(b _{1}+b _{2})\] \[(43080-b _{1}^{2})/b _{1}=b _{2}\]
b1=120 and b2=239 happens to fit that.
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!
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. ;-)
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.
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)
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.
And yes, I understand. Thank you for your help!
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 )