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How do I determine the length of the shorter base of a trapezoid from the longer base length, height, and only two angles?

An example would be 24" longer base, with 45 deg angles at both ends with only 1" in height. Both upper angles would be 135 deg.

What would the length of the shorter base be?

How do you solve for it?

Thanks!

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4 Answers 4

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Suppose the longer base length is $b$ and the $2$ angles are $\alpha$ and $\beta$ with height being $h$

The shorter base length is

$$b-h\cot \alpha-h\cot \beta = b-h(\cot \alpha+ \cot \beta)$$

To see this, notice that the height and the base form perpendicular angle.

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If you decompose the figure into a rectangle and two right triangles, and note the triangles are isosceles, you'll see the length you seek is a longer base minus the height doubled.

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One way to think about it is this: the longer base and the angles determine a triangle. The height determines where the similar triangle is cut off to give the trapezium. Thus, if we let the base you want be $b,$ and the other $B,$ then it follows that $$\frac bB=\frac aA.$$

You can determine $A,$ the height of the main triangle, from the given information. Then $a,$ the height of the cut-off triangle, is given by $a=A-h,$ with $h$ being the height of the trapezium, also given.

The work is in determining $A.$ One way is to first calculate some other side of the large triangle (use the cosine rule, for example -- remember all three angles are known, and one side). Then you now have two sides and an included angle. This allows you to determine $A$ as a scaled sine of the included angle.

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In the specific example:

As the given base angles are equal, the trapezoid is isosceles. The shorter side then is

$$ 24- 1 \cdot \cot 45^{\circ}- 1 \cdot \cot 45^{\circ} =22^{''}$$

General question:

If the base side dimension other than the specific 24^{''}$ is given as asked in the second para of question you asked then we can view it generally to choices when the parallel sides get extended by the same amount:

When the $\pm 45^{\circ}$ slant sides are extended it makes a right angle at intersection i.e., at the vertex.

We can draw parallels making similar right triangles. Not shown.

Between parallels the minimum distance is constant. That is, difference of the longer and shorter sides is constant.

Differences of two parallel sides of an isosceles trapezium so its sides make up a triangle and its height ( between parallel sides) only are given.

Due to insufficient data the trapezium cannot be uniquely constructed, but can be constructed/ solved only upto an arbitrary displacement constant motion of vertex $c$ parallel to longer base as shown.

Shorter side $c$ is indeterminate.

Some trapezium solutions are shown, modified from right triangle of sides $(\sqrt2, \sqrt2,1)$.

So the length of the shorter base of a trapezoid from the longer base length, height, and only two angles cannot be determined.

The following answer sketch applies to your question in the first para:

enter image description here

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  • $\begingroup$ Given is the length of the longer base, not the difference in lengths. $\endgroup$ Commented Nov 6, 2021 at 15:18
  • $\begingroup$ You can choose the length of base as anything, including $24^{''} $in the given specific problem, shorter side being $22^{''},$ the other dimensions are satisfied. $\endgroup$
    – Narasimham
    Commented Nov 6, 2021 at 15:27
  • $\begingroup$ You do not choose what is given. $\endgroup$ Commented Nov 6, 2021 at 15:41
  • $\begingroup$ The first para in OP's question was general and the second case was a particular one. In the first case there was no need to use what is given. I tried to address both aspects. Hope it is in order. $\endgroup$
    – Narasimham
    Commented Nov 6, 2021 at 21:45

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