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

Summary

In this instructional video, the instructor guides viewers through a geometric problem involving the ratio of segment lengths in similar triangles. The focus is on identifying the correct pair of similar triangles among three given options by matching angle measures. Once identified, the instructor explains that the ratio of corresponding sides in similar triangles is constant. The example uses a 35-degree and 90-degree angle to establish similarity and approximates the ratio of segment PN to MN as roughly 0.6, suggesting choice B as the closest answer.

Takeaways

• 📐 The task is to approximate the ratio of the length of segment PN to the length of segment MN.
• 🔍 The solution involves identifying similar triangles among the given options.
• 🔢 Similarity in triangles is determined by having two angles in common, which implies the third angle is also the same.
• 📌 A 35-degree angle and a 90-degree angle are common in the triangles being compared.
• 🔎 Triangle two is identified as having a 35-degree angle, a 90-degree angle, and a 55-degree angle, making it similar to triangle PNM.
• 🌐 The angles in triangle PNM and triangle two add up to 180 degrees, confirming their similarity.
• 📏 The ratio of corresponding sides in similar triangles will be the same.
• 📊 PN corresponds to the side opposite the 35-degree angle, and MN corresponds to the side opposite the 55-degree angle.
• ✂️ The ratio of PN to MN is equivalent to the ratio of the corresponding sides in triangle two, which is 5.7 over 8.2.
• 🧩 It's important to note that the actual lengths of the sides are not 5.7 and 8.2, but the ratio is what is significant.
• 🔑 The final step involves approximating the ratio 5.7/8.2 to find the closest answer choice, which is suggested to be around 0.6.

Q & A

What is the main objective of the video script?

The main objective is to approximate the ratio of the length of segment PN to the length of segment MN using the concept of similar triangles.

What is the significance of the 35-degree angle in the script?

The 35-degree angle is significant because it is a common angle between triangle PNM and one of the given triangles, which helps in identifying the similar triangles.

How does the presence of a 90-degree angle contribute to the problem?

The presence of a 90-degree angle indicates a right triangle, which is one of the criteria for identifying similar triangles in the context of this problem.

What mathematical concept is used to determine the similarity of triangles?

The concept of angle-angle similarity is used, which states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Why is it important to identify similar triangles in this context?

Identifying similar triangles is important because it allows us to establish that the ratios of corresponding sides are equal, which is necessary to approximate the ratio of PN to MN.

What is the ratio of the red side to the blue side in the similar triangles?

The ratio of the red side to the blue side in the similar triangles is 5.7 to 8.2, as these sides correspond to the segments PN and MN, respectively.

How does the instructor approximate the ratio of 5.7 to 8.2?

The instructor approximates the ratio by dividing 57 by 82 and estimating that it is a little less than 0.7, suggesting a value around 0.6.

What is the purpose of the color coding in the script?

The color coding is used to visually distinguish between the corresponding sides of the similar triangles, making it easier to identify which sides are being compared in the ratio.

Why can't the exact lengths of the sides be determined from the ratio alone?

The exact lengths cannot be determined from the ratio alone because similarity only tells us that the ratios of corresponding sides are equal, not the actual lengths of the sides.

What is the conclusion the instructor reaches after approximating the ratio?

The instructor concludes that the approximate ratio is between 0.57 and 1, and after a rough calculation, suggests that it is likely around 0.6, which corresponds to choice B in the given options.

Outlines

00:00

📐 Understanding Similar Triangles and Ratio Approximation

The instructor introduces a problem involving three triangles and the task of approximating the ratio of segment PN to segment MN. The video encourages viewers to pause and attempt the problem first. The instructor then explains that the solution likely involves triangle similarity, which is determined by having two common angles. By identifying the triangles with a 35-degree and a 90-degree angle, the instructor concludes that triangle PNM and the second triangle are similar due to their matching angles. The concept of corresponding sides in similar triangles having equal ratios is used to find the ratio of PN to MN, which is approximated by comparing the sides opposite the 35-degree and 55-degree angles in the similar triangles. The instructor colors the segments for clarity and approximates the ratio to be between 0.6 and 0.7, suggesting choice B as the closest answer.

Keywords

💡Ratio

A ratio is a mathematical expression that compares two quantities by division. In the context of the video, the ratio of PN to MN is the main focus, representing the division of the length of segment PN by the length of segment MN. The video script discusses approximating this ratio as part of solving a geometric problem involving triangles.

💡Triangle

A triangle is a polygon with three edges and three vertices. The video script involves analyzing three different triangles to determine which one can be used to approximate the ratio of segment lengths. Triangles are central to the theme of the video as they are the geometric figures being studied.

💡Similarity

Similarity in geometry refers to a property of shapes where one is a scaled version of the other, with corresponding angles and sides in proportion. The script mentions using similarity to determine which of the three triangles can be used to approximate the ratio, by identifying triangles with two common angles.

💡Angle

An angle is a measure of rotation or the space between two lines or rays that have a common endpoint. The script identifies specific angles within the triangles, such as 35 degrees and 90 degrees, which are crucial in determining the similarity between triangles.

💡Evaluate

To evaluate, in a mathematical context, means to calculate or determine the value of an expression or equation. The video script instructs the viewer to 'evaluate' the ratio, which involves finding the numerical value of PN divided by MN.

💡Corresponding Sides

In similar triangles, corresponding sides are the sides that are in the same relative position in each triangle. The script uses the concept of corresponding sides to establish that the ratio of one set of sides in one triangle will be the same as the ratio of the corresponding sides in another similar triangle.

💡Approximate

To approximate means to estimate a value or quantity that is close to, but not exactly, the exact or actual value. The video script asks the viewer to approximate the ratio of PN to MN, suggesting that an exact value may not be necessary but a close estimate will suffice.

💡Segment

A segment in geometry refers to a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. The script uses the term 'segment' to refer to the parts of the triangles, specifically PN and MN, whose ratio is being sought.

💡Congruence

Congruence in geometry means that two shapes are identical in shape and size. The script contrasts congruence with similarity, noting that while the triangles are similar (same shape, different sizes), they are not congruent, which is why we can't assume exact lengths of the sides but only their ratios.

💡Divide

In mathematics, to divide means to separate a quantity into a number of equal parts. The script uses division to calculate the ratio of the lengths of segments PN and MN, with the process of dividing 5.7 by 8.2 to approximate the ratio.

Highlights

The task is to approximate the ratio of the length of segment PN to the length of segment MN.

The solution involves identifying similar triangles to approximate the ratio.

Similar triangles are identified by having two angles in common.

Triangle PNM and triangle number two are similar due to common angles.

The angles in triangle PNM and triangle number two are 35 degrees, 90 degrees, and 55 degrees.

The ratio of corresponding sides in similar triangles is the same.

The ratio of PN to MN is equivalent to the ratio of the side opposite the 35-degree angle to the side opposite the 55-degree angle.

The specific ratio given is 5.7 over 8.2.

The length of the sides is not necessarily 5.7 and 8.2, but the ratio is.

The ratio is approximated to be between 0.57 and 1.

The ratio is further approximated to be around 0.6.

The method of approximating the ratio involves dividing the numbers by hand.

The division of 8.2 into 5.7 is compared to dividing 82 into 57.

The division results in a number slightly less than 0.7.

The closest choice to the approximated ratio is suggested to be choice B.

The importance of understanding the properties of similar triangles is emphasized.

The practical application of the ratio in geometry problems is demonstrated.

Transcripts

- [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.