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Side ratios in right triangles as a function of the angles

 By similarity, side ratios in right triangles are properties of the angles in the triangle.
When we studied congruence, we claimed that knowing two angle measures and the side length between them (Angle-Side-Angle congruence) was enough for being sure that all of the corresponding pairs of sides and angles were congruent.
How can that be? Even with the Pythagorean theorem, we need two side lengths to find the third. In this article, we'll take the first steps towards understanding how the angle measures and side lengths give us information about each other in the special case of right triangles.
This is a great opportunity to work with a friend or two. The goal of this article is to find and discuss patterns, not to spend a bunch of time calculating. Try splitting up the work so there's more time to talk about what you see!

Let's look for patterns

First, we'll collect some data about a set of triangles.
How are the four triangles related?
The triangles are
according to the
criterion.

Measurement table
Here are those triangles again.
Complete the table of measurements relative to A.
ABCADEAFGAHI
Opposite leg length691215
Adjacent leg length8
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
16
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
Hypotenuse length1015
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
25
Angle A37°37°37°37°
Right angle90°90°90°90°
Last angle
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
°
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
°
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
°
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
°

Now we're ready to start checking that data for patterns.
Ratio table
Complete the ratio table.
Round to the nearest hundredth.
ABCADEAFGAHI
adjacent leg lengthhypotenuse length
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
opposite leg lengthhypotenuse length
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
opposite leg lengthadjacent leg length
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

What did you notice?

Proving that the pattern works for another angle measure

Proof
Complete the proof that ACBC=FDED.
StatementReason
1AFAll right angles are congruent.
2BEGiven
3ABC
similarity
4ACFD=BCEDLengths of corresponding sides of similar triangles form equal ratios.
5ACBC=FDEDMultiply both sides by
.

Conclusion of proof
What did we prove?
Choose 1 answer:
What triangles did we prove it for?
Choose 1 answer:

What did we conclude?

If two right triangles have an acute angle measure in common, they are similar by angle-angle similarity. The ratios of corresponding side lengths within the triangles will be equal. So the ratio of the side lengths of a right triangle just depends on one acute angle measure.

Why will this be useful?

Before, we could use the Pythagorean theorem to find any missing side length of a right triangle when we knew the other two lengths. Now, we have a way to relate angle measures to the right triangle side lengths. That allows us to find both missing side lengths when we only know one length and an acute angle measure. We can even find the acute angle measures in a right triangle based on any two side lengths.
Extension 1.1
Given the measure of an acute angle in a right triangle, we can tell the ratios of the lengths of the triangle's sides relative to that acute angle.
Here are the approximate ratios for angle measures 25°, 35°, and 45°.
Angle25°35°45°
adjacent leg lengthhypotenuse length0.910.820.71
opposite leg lengthhypotenuse length0.420.570.71
opposite leg lengthadjacent leg length0.470.71
Use the table to approximate mJ in the triangle below.
Choose 1 answer:

Want to join the conversation?

  • starky seedling style avatar for user Bailey White
    What the heck? I’m really confused.
    (76 votes)
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    • piceratops ultimate style avatar for user PiFace3.1415
      It's basically saying that if you know one angle of a right angle triangle (other than the 90 degree angle) you can use this to deduce what the ratios of the side lengths are, whether that be opposite / hypotenuse (sin), Adjacent / Hypotenuse (cos), or opposite / adjacent (tan). Once you know the ratios, as soon as you have a side length, you can use SOHCAHTOA to find all of the side lengths, which should be covered in other lessons.
      At the moment all that you really need to know is that all triangles with the same angles will have the same side length ratios.
      (73 votes)
  • blobby green style avatar for user mrsdanielgray
    I think this whole thing is complicated.
    (39 votes)
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  • male robot hal style avatar for user Kasim Ahmed
    I have been using Khan Academy for years! Returning to Khan Academy after a long break to refresh my memory on trigonometry. This trigonometry course is all over the place. When I first studied trigonometry on this site a few years ago, the flow and structure of the unit was clearer and progressed incrementally. In this unit, there are some simple concepts followed by complex concepts. This page needs to be restructured to make it easier for learners, as it was a few years ago!
    (34 votes)
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  • blobby green style avatar for user gamag299th
    What in the congruent relationship of pythagoras and a rectangular prism? These instructions don't teach you. Where do I learn what a side-side-side angle and a angle-angle is??
    (13 votes)
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  • orange juice squid orange style avatar for user shelley
    in Proving that the pattern works for another angle measure, can someone reexplain the last row on the table? i dont understand
    (8 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      It says that if we multiply both sides of the equation in step 4
      (𝐴𝐶∕𝐹𝐷 = 𝐵𝐶∕𝐸𝐷)
      by some factor 𝑘, we end up with the equation in step 5
      (𝐴𝐶∕𝐵𝐶 = 𝐹𝐷∕𝐸𝐷)

      This gives us the system of equations
      𝑘⋅𝐴𝐶∕𝐹𝐷 = 𝐴𝐶∕𝐵𝐶
      𝑘⋅𝐵𝐶∕𝐸𝐷 = 𝐹𝐷∕𝐸𝐷

      We can solve for 𝑘 in either equation.
      In both cases we get 𝑘 = 𝐹𝐷∕𝐵𝐶,
      which is our answer.
      (9 votes)
  • male robot hal style avatar for user caldwelljt
    I can't help but feel this was horribly explained (for how I think)... I know this is just one of those things i'll have to look away and come back to in a couple hours to get the crux of it.
    (7 votes)
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  • blobby green style avatar for user avnish.devekar.438
    I don't understand how they associated Angle measures with the length of the line segment.
    (4 votes)
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  • sneak peak blue style avatar for user Stephen Earley
    So wait I don't quite get it. At the end we conclude that now we have a way to figure out all the side lengths of a triangle if we know one side length and one of the angle measures. I understand up to that point. But I don't know, say if I know these values in a given triangle, what to do from there. I'm just given these pre-completed tables. So are we going to learn how to get these numbers I see in the table later on? Or did I miss something in these lessons?
    Thanks!
    (2 votes)
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    • mr pink green style avatar for user David Severin
      If you include that fact that it has to be a right triangle, then you are correct that having a side and an angle (or two sides) will allow you to find all other parts of the triangle (approximations for the most part). However, we generally find the numbers on the table with use of a calculator, not by hand. There should be trig functions on any graphing calculator. You will progress to non-right triangles later, but you need more information for them.
      (12 votes)
  • blobby purple style avatar for user Anna Liu
    y is this so complicated is it only me?
    (7 votes)
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  • male robot hal style avatar for user San(gharsha)
    What is the difference between "leg" and just the name?
    (4 votes)
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