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## High school geometry

### Unit 5: Lesson 4

Ratios in right triangles# Using similarity to estimate ratio between side lengths

CCSS.Math:

When two right triangles share an acute angle measure, the ratios of the corresponding side lengths within the triangles are equal. Created by Sal Khan.

## Video transcript

- [Instructor] So we've
been given some information about these three triangles here. And then they say, "Use
one of the triangles," so use one of these three triangles, "to approximate the ratio." The ratio's the length
of segment PN divided by the length of segment MN. So they want us to figure
out the ratio of PN over MN. So pause this video and see
if you can figure this out. All right, now let's work
through this together. Now, given that they want
us to figure out this ratio and they want us to actually evaluate it or be able to approximate it, we are probably dealing with similarity. And so what I would wanna look for is, are one of these triangles similar to the triangle we have here? And we're dealing with similar triangles if we have two angles in common. Because if we have two angles in common, then that means that we definitely have the third angle as well,
because the third angle's completely determined by what
the other two angles are. So we have a 35 degree angle here. And we have a 90 degree angle here. And out of all of these choices, this doesn't have a 35
degree angle, it has a 90. This doesn't have 35, has a 90. But triangle two here
has a 35 degree angle, has a 90 degree angle and
has a 55 degree angle. And if you did the math, knowing that 35 plus 90 plus this have to add up to 180 degrees, you would see that this too has a measure of 55 degrees. And so given that all of
our angle measures are the same between triangle PNM and triangle number two right over here, we know that these two
are similar triangles. And so the ratios between
corresponding sides are going to be the same. We could either take the
ratio across triangles. Or we could say the ratio within, where we just look at one triangle. And so if you look at PN over MN, let me try to color code it. So PN, right over here, that corresponds to the side that's opposite the 35 degree angle. So that would correspond to this side, right over here on triangle two. And then MN, that's this that I'm coloring in this blueish color not so well, probably spend more time coloring. That's opposite the 55 degree angle. And so opposite the 55
degree angle would be right over there. Now, since these triangles are similar, the ratio of the red side,
the length of the red side over the length of the blue side is going to be the same in either triangle. So PN, let me write it this way. The length of segment PN over the length of segment MN is going to be equivalent to 5.7 over 8.2. 'Cause this ratio is going to be the same for the corresponding sides, regardless of which triangle you look at. So if you take the side
that's opposite 35 degrees, that's 5.7 over 8.2. Now to be very clear, it
doesn't mean that somehow the length of this side is 5.7 or that the length of this side is 8.2. We would only be able
to make that conclusion if they were congruent. But with similarity, we
know that the ratios, if we look at the ratio of the red side to the blue side on
each of those triangles, that would be the same. And so this gives us that ratio. And let's see, 5.7 over 8.2, which of these choices get close to that? Well, we could say that this is roughly, if I am approximating it, let's see, it's going to be larger than 0.57. Because 8.2 is less than 10. And so we are going to
rule this choice out. And 5.7 is less than 8.2. So it can't be over one. And so we have to think
between these two choices. Well, the simplest thing I can do is actually just try to
start dividing it by hand. So 8.2 goes into 5.7 the same number of times as 82 goes into 57. And I'll add some decimals here. So it doesn't go into 57. But how many times does 82 go into 570? I would assume it's about 6 times, maybe seven times, looks like. So seven times two is 14. And then seven times eight is 56. This is 57. So it's actually a little less than 0.7. This maybe go a little bit too high. So if I am approximating,
it's gonna be 0.6 something. So I like choice B, right over there.