Using similarity to estimate ratio between side lengths | High school geometry | Khan Academy

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 is the length of segment PN divided by the length of segment MN.

So they want us to figure out the ratio 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.

What I would want to look for is, are one of these triangles similar to the triangle we have here? 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 is 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. Of all of these choices, this doesn't have a 35-degree angle; it has a 90.

This doesn't have a 35; it has a 90. But triangle 2 here has a 35-degree angle, has a 90-degree angle, and has a 55-degree angle.

If you did the math knowing that 35 plus 90 plus this have to add up to 180 degrees, you would see that this 2 has a measure of 55 degrees.

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.

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 when we just look at one triangle.

If you look at PN over MN, let me try to color code it. So PN right over here corresponds to the side that's opposite the 35-degree angle.

That would correspond to this side right over here on triangle 2. Then MN, that's this that I'm coloring in this bluish color—not so well. I've probably spent 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 over 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, say the length of segment PN over this length of segment MN is going to be equivalent to 5.7 over 8.2.

Because 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.

This gives us that ratio.

Let's see, 5.7 over 8.2. Which of these choices gets 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.

5.7 is less than 8.2, so it can't be over one. 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.

I would 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 six times, maybe seven times.

Looks like so 7 times 2 is 14, and then 7 times 8 is 56 because it's 57.

So it's actually a little less than 0.7; this got made me go a little bit too high. If I am approximating, it's going to be 0.6 something.

So I like choice B right over there.