Need help constructing a trapezoid

[rocketFrog]

Hi everyone,

I'm doing a problem where I'm supposed to construct a trapezoid given the sum of parallel bases, a height and the two angles on the longer base. Suppose the trapezoid's vertexes are A, B, C and D, and the bases are a and c, where base a is the longer one. The angles given are the angles A and B.

I'm stuck since I'm not sure about how to use the sum of bases to get any side lengths. I thought I could work something out with the midpoint (since the midpoint = sum of bases / 2) but nothing seems to click.

Here's a picture of an example trapezoid with a midpoint:



Any help would be appreciated!

 

[blamocur]

Could you define "construct" please? Straightedge and compass construction? Or computing X,Y coordinates of the vertices?

 

[rocketFrog]

Could you define "construct" please? Straightedge and compass construction? Or computing X,Y coordinates of the vertices?

Straightedge and compass. Forgot to mention.

 

[Dr.Peterson]

Hi everyone,

I'm doing a problem where I'm supposed to construct a trapezoid given the sum of parallel bases, a height and the two angles on the longer base. Suppose the trapezoid's vertexes are A, B, C and D, and the bases are a and c, where base a is the longer one. The angles given are the angles A and B.

I'm stuck since I'm not sure about how to use the sum of bases to get any side lengths. I thought I could work something out with the midpoint (since the midpoint = sum of bases / 2) but nothing seems to click.

I will suppose that "construct" means essentially to "solve", that is, to find all sides (and angles) so that it could be drawn.

From the height and angles A and B, you can use simple trigonometry to the lengths d and b. Similarly, you can get the difference of the bases, and use that and the sum of bases to find the bases a and c themselves.

You don't appear to have used the angles at all. Do you know basic trigonometry?

By the way, you appear to be confusing "midpoint" with "midline". And your "(a-c)/2" is wrong; that would be true only if it were an isosceles trapezoid. But the sum of that segment AE and the corresponding BF is a-c.

 

[The Highlander]

Hi everyone,

I'm doing a problem where I'm supposed to construct a trapezoid given the sum of parallel bases, a height and the two angles on the longer base. Suppose the trapezoid's vertexes are A, B, C and D, and the bases are a and c, where base a is the longer one. The angles given are the angles A and B.

I'm stuck since I'm not sure about how to use the sum of bases to get any side lengths. I thought I could work something out with the midpoint (since the midpoint = sum of bases / 2) but nothing seems to click.

Here's a picture of an example trapezoid with a midpoint:

View attachment 33750

Any help would be appreciated!

You say that the angles A and B are "given" but you don't tell us what they are!
The length AE also appears to be "given" as: \(\displaystyle \frac{a-c}{2}\). Is that correct (ie: another "given" in the original question)? Or is it something you have 'added' as part of your attempt(s) to solve the problem?
If you don't provide full information then it is very difficult for anyone to offer sensible assistance.
A response to blamocur's query is also necessary.

 

[rocketFrog]

Thanks for the replies!

I mean just finding a way to practically construct a trapezoid, using a compass and ruler. The angles and side lengths are totally arbitrary. I just need a practical strategy to construct it, you know, divide it into manageable steps.

 

[Dr.Peterson]

Straightedge and compass. Forgot to mention.

Showing us a diagram with coordinates didn't help communicate that!

Now please show us an attempt at the kind of construction you are supposed to do. I suppose you must be given the height and sum of bases as line segments, and the angles as actual angles, each of which you can copy as needed?

It may be helpful to show us an image of the problem as given to you.

 

[rocketFrog]

Here's an example.

I only put vertex A on the line l - as you can see in steps, to prevent causing further confusion since I'm not sure what further steps to take. But yeah, what I'm supposed to do from here is construct the angles, use the side lengths and somehow get a trapezoid. That's what it's all about.

Maybe important mentioning this is based on triangle construction? I tried dividing the trapezoid into triangles with 'constructible' elements using the height but failed.

 

[The Highlander]

Thanks for the replies!

I mean just finding a way to practically construct a trapezoid, using a compass and ruler. The angles and side lengths are totally arbitrary. I just need a practical strategy to construct it, you know, divide it into manageable steps.

Again, you said, initially, that the height, the sum of the bases and the two angles were "given"!

Hi everyone,

I'm doing a problem where I'm supposed to construct a trapezoid given the sum of parallel bases, a height and the two angles on the longer base.

So, are they "given" or just "totally arbitrary", as you now seem to be saying?

Please show the complete original question.

 

[Dr.Peterson]

Here's an example.

I only put vertex A on the line l - as you can see in steps, to prevent causing further confusion since I'm not sure what further steps to take. But yeah, what I'm supposed to do from here is construct the angles, use the side lengths and somehow get a trapezoid. That's what it's all about.

Maybe important mentioning this is based on triangle construction? I tried dividing the trapezoid into triangles with 'constructible' elements using the height but failed.

View attachment 33751

I too would like to see an image of the original problem, just to be sure we (and you) understand it correctly; but it appears to be as I expected, that you are given two segments and two drawn angles (which could be arbitrary as far as the poser of the problem is concerned!), and need to use those for the construction.

Now please show a couple steps you've tried beyond placing the point A. I would expect you to copy angle A, and perhaps to construct a parallel line at distance h. Do you know ways to do those?

When you say, "this is based on triangle construction", do you mean that is a topic you have recently learned, or that you have tried a method that uses such a construction, but didn't show it to us? The steps I suggested just now amount to constructing a right triangle given an angle and a side.

 

[rocketFrog]

I too would like to see an image of the original problem, just to be sure we (and you) understand it correctly; but it appears to be as I expected, that you are given two segments and two drawn angles (which could be arbitrary as far as the poser of the problem is concerned!), and need to use those for the construction.

Now please show a couple steps you've tried beyond placing the point A. I would expect you to copy angle A, and perhaps to construct a parallel line at distance h. Do you know ways to do those?

When you say, "this is based on triangle construction", do you mean that is a topic you have recently learned, or that you have tried a method that uses such a construction, but didn't show it to us? The steps I suggested just now amount to constructing a right triangle given an angle and a side.

Yes, exactly!
This is the original problem:

OIV5. Construct a trapezoid if the sum of the bases, the height of the trapezoid and the angles are given on a larger base.

I drew angle A, the height, drew line r parallel to line l. Vertex D is where line q (line of angle A) and line r (the parallel line) intersect. But I'm pretty much stuck here since I don't know what to do with angle B or the sum of bases.

This isn't a topic I've recently learned, this is a topic I haven't learned well enough, trying to fill in the gaps right now. I've constructed triangles before, given for example a base and it's two angles - this is one of the simpler examples. But I've never constructed a trapezoid.

 

[Dr.Peterson]

OIV5. Construct a trapezoid if the sum of the bases, the height of the trapezoid and the angles are given on a larger base.

I'm curious: What is OIV5? And what is the context of the problem? Is this perhaps preparation for some contest, or exam, or something? Knowing your context can be very helpful.

I assume you know how to construct EF so that it is perpendicular, and r so that it is parallel, and so on, and you are just summarizing the general steps.

I drew angle A, the height, drew line r parallel to line l. Vertex D is where line q (line of angle A) and line r (the parallel line) intersect. But I'm pretty much stuck here since I don't know what to do with angle B or the sum of bases.

Now you need to figure out how to choose the base length. One trick I find useful in this sort of problem is to suppose you have finished the construction (somehow), and look back at what you have. So draw a trapezoid and assume that the sum of the bases is what you were given:

​

ABCD is what you want in the end; AE is the sum of the bases. (Observe that BE = CD.) I also drew in EF with the angle you want to have at B, because that seems like it might be useful. Are there any relationships you can see such that you might be able to find point B, having drawn the rest first?

I've constructed triangles before, given for example a base and it's two angles - this is one of the simpler examples. But I've never constructed a trapezoid.

Inventing a construction is a challenging task, similar to writing proofs; what we're working on here is how to find your way in unfamiliar territory, which is what all problem-solving ultimately is. Don't let the fact that you haven't done this before stop you; explore various possibilities, expecting to have to abandon one way and try another! When you don't know what to do ... that's when it gets fun.

 

[rocketFrog]

: What is OIV5? And what is the context of the problem? Is this perhaps preparation for some contest, or exam, or something? Knowing your context can be very helpful.

O - just a letter to denote all the geometric construction problems
IV - means it's level IV out of V difficulty
5 - the 5th problem on the list

This is from a list of problems my teacher gave me on request to revise during summer break after 10th grade. I have no idea where did these come from and what's the weird OIV5 notation but it's a great set of problems. By the way, this is the only construction problem I got. It's an odd one.

ABCD is what you want in the end; AE is the sum of the bases. (Observe that BE = CD.) I also drew in EF with the angle you want to have at B, because that seems like it might be useful. Are there any relationships you can see such that you might be able to find point B, having drawn the rest first?

Doesn't that put it this way:
- BEFC is a paralellogram
- the angle opposite to angle E and angle E are equal
- the opposite sides are equal, meaning the upper segment DF = 2c

If I'm not wrong which but I very well could be - we could get c by constructing a bisector of DF?

 

[Dr.Peterson]

Doesn't that put it this way:
- BEFC is a paralellogram
- the angle opposite to angle E and angle E are equal
- the opposite sides are equal, meaning the upper segment DF = 2c

If I'm not wrong which but I very well could be - we could get c by constructing a bisector of DF?

I think you've got it.

I wanted to give you a chance to discover, with just a little nudge, and it seems to have worked.

 

[rocketFrog]

I think you've got it.

I wanted to give you a chance to discover, with just a little nudge, and it seems to have worked.

I really appreciate the help Dr. Peterson!