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.