Challenging similarity problem | Similarity | Geometry | Khan Academy

Summary

This script explores the use of similar triangles to determine the length of CF in a geometric diagram. By proving the similarity of triangles ABE and CFE, and DEB and CFB, and setting up equations based on the ratios of corresponding sides, the solution finds CF to be 36/7 units, highlighting a fascinating property of similar triangles.

Takeaways

• 📐 The problem involves determining the length of CF in a geometric diagram involving similar triangles.
• 🔍 Triangle CFE is similar to triangle ABE, which is inferred from their shared angles and embedded geometric structure.
• 🔍 Triangle CFB is also similar to triangle DEB, based on their corresponding angles and parallel lines.
• 📚 To prove similarity, the script demonstrates that corresponding angles in the triangles are congruent, fulfilling the angle-angle (AA) criterion for triangle similarity.
• 🔢 The script introduces variables: BE is assumed to be equal to y, and BF is denoted as x, with FE being y - x.
• 🔄 The ratios of corresponding sides in similar triangles are used to set up equations, leveraging the properties of similar triangles.
• 🧩 The script establishes that the ratio of CF to 9 (from triangle CFE) is equal to the ratio of y - x to y, and similarly, the ratio of CF to 12 (from triangle CFB) is equal to the ratio of x to y.
• 🔎 A key insight is that the ratio of CF to 9 can be rewritten as 1 - (CF/12), allowing for a substitution that simplifies the problem.
• ⚖️ The script solves for CF by combining the equations and finding a common denominator, ultimately leading to the equation 4CF + 3CF = 36.
• 📉 The final solution is CF = 36/7, demonstrating a universal height for the intersection point of strings draped from the tops of two objects to the base of the other, regardless of their distance apart.
• 🌐 The problem highlights a geometric principle where the height of the intersection point remains constant (36/7) for any given heights of 9 and 12, illustrating a fascinating property of similar triangles.

Q & A

What is the main problem discussed in the script?

The main problem discussed in the script is to determine the length of CF in a geometric diagram involving similar triangles.

Why are triangles CFE and ABE considered similar?

Triangles CFE and ABE are considered similar because they both have a 90-degree angle and share another angle, angle CEF, which is congruent to angle AEB.

How can you prove that triangles CFB and DEB are similar?

Triangles CFB and DEB are similar because they both have a 90-degree angle and share another angle, angle DBE, which is congruent to angle CBF.

What is the significance of the angle shared by triangles CFE and ABE?

The shared angle between triangles CFE and ABE is significant because it helps establish that the triangles are similar, which is a key step in solving the problem.

What assumption is made about the length of BE in the script?

The script assumes that the length of BE is equal to y, which is a variable used to help solve the problem.

Why is it useful to assume that BE is equal to y?

Assuming BE is equal to y is useful because BE is shared by both triangles ABE and DEB, and this assumption provides a starting point for setting up equations.

What variables are introduced for the lengths of BF and FE?

The script introduces x as the variable for the length of BF and y - x for the length of FE.

How are the ratios of corresponding sides in similar triangles used in the script?

The ratios of corresponding sides in similar triangles are used to set up equations that relate CF to the other sides of the triangles, ultimately helping to solve for CF.

What is the final equation derived to solve for CF?

The final equation derived to solve for CF is CF/9 + CF/12 = 1, which simplifies to 7CF/36 = 1.

What is the final value of CF determined in the script?

The final value of CF determined in the script is 36/7, or 5 and 1/7, regardless of the distance between the two triangles.

What insight does the problem provide about the intersection of strings draped from the tops of two objects?

The problem shows that regardless of the distance between two objects, the point where the strings intersect will always be at a height of 36/7 from the base, illustrating a consistent geometric property.

Outlines

00:00

📐 Identifying Similar Triangles

The paragraph discusses the task of determining the length of CF in a geometric diagram. The speaker suggests that the problem involves similar triangles, specifically triangles CFE and ABE, and triangles CFB and DEB. The approach involves proving the similarity of these triangles by identifying congruent angles and parallel lines. The speaker also introduces the idea of using ratios of corresponding sides to solve for the unknown length CF.

05:04

🔍 Solving for CF Using Similar Triangle Ratios

This paragraph delves into the mathematical process of solving for the length of CF using the properties of similar triangles. The speaker assigns variables to unknown sides (BE as y and BF as x) and sets up equations based on the ratios of corresponding sides in the triangles. The equations are manipulated to isolate CF, leading to a single equation with one unknown. Through algebraic manipulation, the speaker solves for CF, demonstrating that regardless of the distance between the two triangles, the intersection point of the strings will always be 36/7 units high.

Keywords

💡Similar Triangles

Similar triangles are triangles that have the same shape but may differ in size. They have all corresponding angles congruent and the ratios of their corresponding sides are equal. In the video, the concept is used to establish relationships between different triangles in the diagram, specifically triangles CFE and ABE, and triangles CFB and DEB. The script mentions proving that these triangles are similar by showing that they share corresponding angles, which is key to solving the problem of finding the length of CF.

💡Corresponding Angles

Corresponding angles are angles that occupy the same relative position in two similar figures. In the context of the video, the script discusses proving that triangles are similar by showing that they have corresponding angles that are congruent. For example, angle CEF is the same as angle AEB, and angle DBE is the same as angle CBF, which helps establish the similarity between the triangles.

💡Proportional Sides

Proportional sides in similar triangles refer to the property that the ratios of the lengths of corresponding sides are equal. The video uses this property to set up equations involving the sides of the triangles. For instance, the ratio of CF to its corresponding side in triangle CFE is equal to the ratio of y - x to y, where y is the length of BE and x is the length of BF.

💡90-Degree Angle

A 90-degree angle is a right angle, which is crucial in the video for identifying right triangles and using properties of similar triangles. The script mentions that triangles ABE and CFE both have a 90-degree angle, which is a key factor in proving their similarity. Additionally, the presence of a 90-degree angle in triangles CFB and DEB is used to establish their similarity.

💡Parallel Lines

Parallel lines are lines in a plane that do not meet, no matter how far they are extended. In the video, the script uses the concept of parallel lines to establish that certain angles are congruent, which is a strategy for proving the similarity of triangles. For example, the script mentions that because two angles are the same, the lines forming these angles are parallel, which helps in deducing the similarity of triangles CFE and ABE.

💡Variables

Variables are symbols used to represent unknown quantities in mathematical problems. In the video, variables like y for the length of BE and x for the length of BF are introduced to help solve the problem. These variables are essential in setting up equations based on the properties of similar triangles and ultimately finding the length of CF.

💡Equations

Equations are mathematical statements that assert the equality of two expressions. In the video, equations are derived from the properties of similar triangles and the proportional sides. For example, the script sets up the equation CF/9 = (y - x)/y, which is used to solve for the unknown length CF.

💡Common Denominator

A common denominator is a number that is a multiple of the denominators of two or more fractions. In the video, the script uses the concept of a common denominator to combine fractions and solve for CF. The script multiplies both sides of the equation by 36 (a common denominator for 9 and 12) to simplify the equation and solve for CF.

💡Reciprocal

The reciprocal of a number is 1 divided by that number. In the video, the script uses the reciprocal to solve the equation involving CF. By multiplying both sides of the equation by the reciprocal of 7/36, the script isolates CF and finds its value.

💡Drape a String

In the video, the script uses the metaphor of draping a string from the top of one triangle to the base of another to illustrate the problem. This visual helps explain the geometric relationships and the solution to finding the height where the strings would intersect, which is represented by the length of CF.

💡Units

Units are used to express the magnitude of physical quantities. The script mentions that the solution for the length of CF is independent of the units used, whether it's feet, yards, or meters. This highlights the generality of the solution and its application in various contexts.

Highlights

Introduction to the problem of finding the length of CF using similar triangles.

Assumption that triangle CFE is similar to triangle ABE based on visual intuition.

Assumption that triangle CFB is similar to triangle DEB based on visual intuition.

Proof of similarity between triangles ABE and CFE by showing two corresponding angles are congruent.

Proof of similarity between triangles DEB and CFB by showing two corresponding angles are congruent.

Introduction of variable y to represent the length of BE, shared by triangles ABE and DEB.

Introduction of variable x to represent the length of BF, and y - x for FE.

Use of proportionality in similar triangles to set up equations relating CF to y and x.

Equation setup for CF over 9 equals y - x over y.

Equation setup for CF over 12 equals x over y.

Substitution of x over y with CF over 12 in the equation for CF over 9.

Simplification of the equation to CF over 9 equals 1 minus CF over 12.

Solving the equation by combining terms and finding a common denominator.

Final calculation of CF as 36 over 7, demonstrating a consistent result regardless of the distance between the triangles.

Practical application of the problem, showing that the intersection point of strings draped from the tops of two objects to the base of the other will always be 36/7 high, regardless of the distance.

Emphasis on the unique property of similar triangles in this problem, highlighting the consistent height of the intersection point.

Transcripts

So given this diagram, we need to figure out

what the length of CF right over here is.

And you might already guess that this

will have to do something with similar triangles,

because at least it looks that triangle CFE is similar to ABE.

And the intuition there is it's kind of embedded inside of it,

and we're going to prove that to ourselves.

And it also looks like triangle CFB

is going to be similar to triangle DEB,

but once again, we're going to prove that to ourselves.

And then maybe we can deal with all the ratios

of the different sides to CF right over here

and then actually figure out what CF is going to be.

So first, let's prove to ourselves that these definitely

are similar triangles.

So you have this 90-degree angle in ABE,

and then you have this 90-degree angle in CFE.

If we can prove just one other angle or one other set

of corresponding angles is congruent in both,

then we've proved that they're similar.

And there we can either show that, look, they both share

this angle right over here.

Angle CEF is the same as angle AEB.

So we've shown two corresponding angles in these triangles.

This is an angle in both triangles.

They are congruent, so the triangles

are going to be similar.

You can also show that this line is parallel to this line,

because obviously these two angles are the same.

And so these angles will also be the same.

So they're definitely similar triangles,

so let's just write that down.

Get that out of the way.

We know that triangle ABE is similar to triangle CFE.

And you want to make sure you get it in the right order.

F is where the 90-degree angle is.

B is where the 90-degree angle is,

and then E is where this orange angle is.

So CFE.

It's similar to triangle CFE.

Now let's see if we can figure out that same statement going

the other way, looking at triangle DEB.

So once again, you have a 90-degree angle here.

If this is 90, then this is definitely

going to be 90, as well.

You have a 90-degree angle here at CFB.

You have a 90-degree angle at DEF or DEB,

however you want to call it.

So they have one set of corresponding angles

that are congruent, and then you'll

also see that they both share this angle right over here

on the smaller triangle.

So I'm now looking at this triangle right over here,

as opposed to the one on the right.

So they both share this angle right over here.

Angle DBE is the same as angle CBF.

So I've shown you already that we

have this angle is congruent to this angle,

and we have this angle is a part of both.

So it's obviously congruent to itself.

So we have two corresponding angles

that are congruent to each other.

So we know that this larger triangle over here

is similar to this smaller triangle over there.

So let me write this down.

Scroll over to the right a little bit.

We also know that triangle DEB is similar to triangle CFB.

Now, what can we do from here?

Well, we know that the ratios of corresponding sides,

for each of those similar triangles,

they're going to have to be the same.

But we only have one side of one of the triangles.

So in the case of ABE and CFE, we've only been given one side.

In the case of DEB and CFB, we've

only been given one side right over here,

so there doesn't seem to be a lot to work with.

And this is why this is a slightly more challenging

problem here.

Let's just go ahead and see if we can assume one of the sides,

and actually, maybe a side that's

shared by both of these larger triangles.

And then maybe things will work out from there.

So let's just assume that this length right over here--

let's just assume that BE is equal to y.

So let me just write this down.

This whole length is going to be equal to y,

because this at least gives us something to work with.

And y is shared by both ABE and DEB, so that seems useful.

And then we're going to have to think about the smaller

triangles right over there.

So maybe we'll call BF x.

Let's call BF x.

And then let's FE-- well, if this

is x, then this is y minus x.

So we've introduced a bunch of variables here.

But maybe with all the proportionalities and things

maybe, just maybe, things will work out.

Or at least we'll have a little bit more sense

of where we can go with this actual problem.

But now we can start dealing with the similar triangles.

For example, so we want to figure out what CF is.

We now know that for these two triangles right here,

the ratio of the corresponding sides are going to be constant.

So for example, the ratio between CF and 9,

their corresponding sides, has got

to be equal to the ratio between y

minus x-- that's that side right there--

and the corresponding side of the larger triangle.

Well, the corresponding side of the larger triangle

is this entire length.

And that entire length right over there is y.

So it's equal to y minus x over y.

So we could simplify this a little bit.

Well, I'll hold off for a second.

Let's see if we can do something similar

with this thing on the right.

So once again, we have CF, its corresponding side on DEB.

So now we're looking at the triangle CFB, not

looking at triangle CFE anymore.

So now when we're looking at this triangle,

CF corresponds to DE.

So we have CF over DE is going to be equal to x-- let me

do that in a different color.

I'm using all my colors.

It's going to be equal to x over this entire base

right over here, so this entire BE, which once again, we know

is y.

So over y.

And now this looks interesting, because it

looks like we have three unknowns.

Sorry, we know what DE is already.

This is 12.

I could have written CF over 12.

The ratio between CF and 12 is going

to be the ratio between x and y.

So we have three unknowns and only two equations,

so it seems hard to solve at first,

because there's one unknown, another unknown,

another unknown, another unknown, another unknown,

and another unknown.

But it looks like I can write this right here,

this expression, in terms of x over y,

and then we could do a substitution.

So that's why this was a little tricky.

So this one right here-- let me do it in that same green color.

We can rewrite it as CF over 9 is equal to y minus x over y.

It's the same thing as y over y minus x over y,

or 1 minus x over y.

All I did is, I essentially, I guess you could say,

distributed the 1 over y times both of these terms.

So y over y minus x over y, or 1 minus x minus y.

And this is useful, because we already know what x over y

is equal to.

We already know that x over y is equal to CF over 12.

So this right over here, I can replace with this, CF over 12.

So then we get-- this is the home stretch here-- CF, which

is what we care about, CF over 9 is equal to 1 minus CF over 12.

And now we have one equation with one unknown,

and we should be able to solve this right over here.

So we could add CF over 12 to both sides,

so you have CF over 9 plus CF over 12 is equal to 1.

We just have to find a common denominator here,

and I think 36 will do the trick.

So 9 times 4 is 36, so if you have to multiply 9 times 4,

you have to multiply CF times 4.

So you have 4CF.

4CF over 36 is the same thing as CF over 9,

and then plus CF over 12 is the same thing as 3CF over 36.

And this is going to be equal to 1.

And then we are left with 4CF plus 3CF is 7CF over 36

is equal to 1.

And then to solve for CF, we can multiply

both sides times the reciprocal of 7 over 36.

So 36 over 7.

Multiply both sides times that, 36 over 7.

This side, things cancel out.

And we are left with-- we get our drum roll now.

So all of this stuff cancels out.

CF is equal to 1 times 36 over 7, or just 36 over 7.

And this was a pretty cool problem,

because what it shows you is if you have two things--

let's say this thing is some type of a pole or a stick

or maybe the wall of a building, or who knows what it is--

if this is 9 feet tall or 9 yards tall or 9 meters tall,

and this over here, this other one,

is 12 meters tall or 12 yards or whatever units you want to use,

if you were to drape a string from either of them to the base

of the other, from the top of one of them to the base

of the other--

regardless of how far apart these two things

are going to be-- we just decided they're y apart.

Regardless of how far apart they are,

the place where those two strings would intersect

are going to be 36/7 high, or 5 and 1/7

high, regardless of how far they are.

So I think that was a pretty cool problem.