GCSE Maths - Pythagoras' Theorem And How To Use It #120

Summary

This educational video script teaches viewers how to apply Pythagoras' theorem to find missing side lengths in right-angled triangles. It emphasizes the importance of identifying the right triangle, knowing two sides, and recognizing the hypotenuse. The script walks through step-by-step examples, demonstrating how to label sides as 'a', 'b', and 'c', and then use the formula a^2 + b^2 = c^2 to solve for the unknown side. Practical examples with calculations and the use of a calculator are included to clarify the process, making the concept accessible for learners.

Takeaways

• 📐 To use Pythagoras' theorem, the triangle must be a right-angled triangle with one 90-degree angle.
• 🔍 You need to know the lengths of two sides to apply the theorem; it doesn't matter which two, but there must be one missing length.
• ✅ The Pythagorean equation is a^2 + b^2 = c^2, where c is the hypotenuse (the longest side) and a and b are the other two sides.
• 📝 It's important to memorize the equation as it's frequently used in mathematics.
• 🔄 The order of a and b doesn't matter, but c must be the hypotenuse opposite the right angle.
• 🧮 Substitute the known side lengths into the equation and solve for the unknown side.
• 📉 When solving, simplify the equation and use mathematical operations to isolate the unknown side.
• 📏 For example, if sides are 3 and 4 units, and you're solving for the hypotenuse, 3^2 + 4^2 = c^2 simplifies to 9 + 16 = c^2, and c = √25 = 5 units.
• 📘 In exam questions, the sides are often labeled with letters, and you should label them as a, b, and c for clarity.
• 🔢 Use a calculator to find the square and square root when dealing with decimal values or to simplify the calculation process.

Q & A

What is the main topic of the video?

The main topic of the video is how to use Pythagoras' theorem to calculate the missing length of a triangle.

What is the first condition to use Pythagoras' theorem as mentioned in the video?

The first condition to use Pythagoras' theorem is that the triangle must have a right angle, which is 90 degrees.

Which sides of a right-angled triangle are referred to as 'a' and 'b' in Pythagoras' theorem?

In Pythagoras' theorem, the sides that are not the hypotenuse (the longest side opposite the right angle) are referred to as 'a' and 'b'.

What is the hypotenuse of a right-angled triangle called in the context of Pythagoras' theorem?

In the context of Pythagoras' theorem, the hypotenuse of a right-angled triangle is referred to as 'c'.

What is the formula of Pythagoras' theorem?

The formula of Pythagoras' theorem is a squared plus b squared equals c squared, where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse.

How do you determine which side is the hypotenuse when applying Pythagoras' theorem?

In applying Pythagoras' theorem, the hypotenuse is determined by identifying the longest side of the triangle, which is always opposite the right angle.

What is the significance of the equation 'a squared plus b squared equals c squared' in the video?

The equation 'a squared plus b squared equals c squared' is significant as it is the core formula of Pythagoras' theorem, which is used to find the missing side of a right-angled triangle.

Can you use Pythagoras' theorem if two sides of a triangle are known but not the hypotenuse?

Yes, you can use Pythagoras' theorem if two sides of a triangle are known, and you can determine the hypotenuse or the missing side using the theorem.

What is the process to solve for the missing side using Pythagoras' theorem as described in the video?

The process involves labeling the sides as 'a', 'b', and 'c', plugging the known side lengths into the Pythagorean equation, and solving for the missing side by performing algebraic operations and taking the square root.

How does the video demonstrate solving for the missing side of a triangle with given side lengths of 1.7 and 3.2?

The video demonstrates solving for the missing side by squaring the given side lengths, adding them together, and then taking the square root of the sum to find the length of the hypotenuse.

What is the final step to find the length of the missing side after setting up the Pythagorean equation?

The final step is to take the square root of the sum of the squares of the known sides to find the length of the missing side.

Outlines

00:00

📐 Introduction to Pythagoras' Theorem

This paragraph introduces the use of Pythagoras' theorem for calculating the missing side of a right-angled triangle. It emphasizes the need for a 90-degree angle and knowledge of the lengths of two sides to apply the theorem. The equation a² + b² = c² is introduced, where c represents the hypotenuse. The paragraph demonstrates how to label the sides and substitute the known values into the equation to solve for the unknown side, using the example of a triangle with sides of lengths 3 and 4, where the length of the hypotenuse (x) is calculated to be 5.

05:02

🔢 Applying Pythagoras' Theorem to More Complex Problems

The second paragraph delves into applying Pythagoras' theorem to more complex problems involving decimal values. It guides through the process of labeling the sides, setting up the equation, and solving for the unknown side. An example with sides of 1.7 and 3.2 is used to demonstrate the calculation, resulting in an unknown side (x) of 3.62 centimeters. The paragraph also clarifies common conventions in exam questions, such as labeling corners and referring to lines by connected corners, and how to ignore these when setting up the Pythagorean equation.

📏 Advanced Pythagorean Calculations

The final paragraph showcases the application of Pythagoras' theorem in a more advanced context, where the goal is to find the length of a specific side in a triangle with labeled vertices. It illustrates the process of ignoring the given vertex labels and focusing on the sides relevant to the Pythagorean theorem. An example with sides of 5.6 and 10.5 is used, leading to the calculation of the hypotenuse (c) as 11.9 centimeters. The paragraph reinforces the method of labeling and solving the equation, emphasizing the importance of understanding the geometric relationships in the problem.

Keywords

💡Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is central to the video's theme as it is used to calculate the missing side lengths in various triangles. For instance, the script uses the theorem to find the length of 'x' in a triangle where two sides are known.

💡Right-Angle Triangle

A right-angle triangle is a specific type of triangle where one of the angles is exactly 90 degrees. The video emphasizes that the Pythagorean Theorem can only be applied to right-angle triangles. The script uses this concept to explain when and how to apply the theorem, highlighting that the presence of a 90-degree angle is a prerequisite for using the theorem.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, which is always opposite the right angle. In the context of the video, the hypotenuse is labeled as 'c' and is the side that is often calculated using the Pythagorean Theorem when the lengths of the other two sides ('a' and 'b') are known. The script mentions that 'c' is always opposite the right angle, and it is crucial for setting up the equation correctly.

💡Sides a and b

In the Pythagorean Theorem, 'a' and 'b' represent the two shorter sides of a right-angled triangle, which are adjacent to the right angle. The video script uses these terms to demonstrate how to set up the theorem's equation, emphasizing that the order of 'a' and 'b' does not matter, but they must be the two sides that are not the hypotenuse. The script provides examples where 'a' and 'b' are substituted with the known side lengths to solve for the unknown side.

💡Equation

The term 'equation' in the video refers to the mathematical statement that uses the Pythagorean Theorem to express the relationship between the sides of a right-angled triangle. The script explains that the equation 'a squared plus b squared equals c squared' is essential for solving problems involving right-angled triangles. It is used multiple times throughout the video to calculate missing side lengths.

💡Calculation

Calculation, as used in the video, refers to the process of performing mathematical operations to solve for an unknown value using the Pythagorean Theorem. The script walks through several examples of calculations, such as squaring the known side lengths, adding them together, and then finding the square root to determine the length of the hypotenuse or one of the other sides.

💡Square Root

The square root operation is mentioned in the video as a method to solve for the unknown side length after setting up the equation from the Pythagorean Theorem. The script explains that once you have 'a squared plus b squared equals c squared', you take the square root of both sides to isolate 'c' and find its value, which is the length of the unknown side.

💡Unknown Side

An unknown side in the context of the video is a side length in a right-angled triangle that needs to be determined using the Pythagorean Theorem. The script uses terms like 'x' and 'c' to represent these unknowns and demonstrates how to calculate their values once the lengths of the other two sides are known. This concept is central to the video's educational purpose, which is to teach viewers how to find missing side lengths.

💡Labeling Sides

Labeling sides is a crucial step highlighted in the video before applying the Pythagorean Theorem. The script instructs viewers to label the sides of the triangle as 'a', 'b', and 'c', with 'c' being the hypotenuse. This labeling is essential for correctly setting up the equation and solving for the unknown side. The video script provides examples of how to label sides and emphasizes the importance of this step in the problem-solving process.

💡Significant Figures

Significant figures are mentioned in the video in relation to the precision required when presenting the final calculated values. The script explains that when solving for an unknown side, such as 'x', and the result is a decimal, it is important to round to the correct number of significant figures as requested in the problem. This concept is important for ensuring that the calculated answers are presented with the appropriate level of precision.

Highlights

Pythagoras' theorem is used to calculate the missing length of a right-angled triangle.

The triangle must have a 90-degree angle to apply Pythagoras' theorem.

You need to know the lengths of two sides to use Pythagoras' theorem.

There should be one missing length that you are trying to find.

The Pythagorean equation is a^2 + b^2 = c^2, where c is the hypotenuse.

The order of a and b does not matter, but c must be the hypotenuse.

To solve for c, substitute the known values into the equation and simplify.

Square root both sides of the equation to isolate c.

For the example with sides 3 and 4, the missing side c is calculated to be 5.

In problems with decimal side lengths, use a calculator to compute the squares and square roots.

When solving, ensure to keep the significant figures as required by the question.

Examined questions often label the corners, and you should label your sides as a, b, and c accordingly.

Ignore the given corner labels and focus on labeling your sides for the Pythagorean equation.

In the example with sides 5.6 and 10.5, the missing side c is found to be 11.9 centimeters.

For the last example with sides 8 and 11, the missing side ac is calculated to be 13.6 centimeters.

The video provides a step-by-step guide on how to apply Pythagoras' theorem to various triangle problems.

The video concludes with a call to like, subscribe, and visit the website for more information.

Transcripts

[Music]

in this video we're going to cover how

we can use pythagoras's theorem to

calculate the missing length of a

triangle

like in this one here where we're trying

to find the length of x

now the first thing we need to look at

is which questions you should actually

use pythagoras for in the first place

first we need to be looking at a right

angle triangle

so if the triangle doesn't have a 90

degree angle like this one

then we can't use pythagoras to solve it

and we'd have to use some other method

you also need to know the lengths of two

of the sides

it doesn't matter which two you know but

you do have to know two of them

like in this one where we know the

length of three and four

and lastly there has to be one missing

length that we're trying to find

so x in this case

the equation for pythagoras theorem is

that a squared plus b squared equals c

squared

and it's really important that you

memorize this equation because it comes

up quite a lot

letters a b and c refer to the three

sides of a triangle

c is always the longest side of the

triangle which we call the hypotenuse

and it will always be opposite the right

angle

we've labeled c the other two sides will

always be a and b

and it doesn't matter which way round

you put them

so we could have a on the left and b on

the top like we have here

or we could swap them over

it doesn't really matter

as long as we have c as our hypotenuse

and the other two sides are a and b

then we're fine

to use the equation all we need to do is

plug the numbers that are corresponding

to each letter into the equation

so because a is four and b is three we'd

write four squared plus three squared

equals c squared

and to work out what c is all we need to

do is solve it like we would with any

other equation in maths

so we can simplify it to 16 plus 9

equals c squared

and then 25 equals c squared

and to get c by itself we just need to

square root both sides

which leaves us with five equals c

so the length of x must be five

have a go at doing the same thing for

this question

so again we're trying to work out the

length of an unknown side x

just like before the first step is to

label our sides so a b and c

and in this case the unknown side is c

because that's our longest length

next we need to write out our equation

so a squared plus b squared equals c

squared

and then we can plug in our values from

the triangle

so 1.7 squared

plus 3.2 squared equals c squared

or instead of writing c squared

we could put x squared because we're

trying to work out x so that's

effectively c in this equation

then the final step is to solve the

equation

a question like this will normally be a

calculator paper

so the best thing to do is put the whole

1.7 squared plus 3.2 squared into the

calculator in one go

which gives us 13.13

which has to equal x squared

so to find x we need to square root both

sides

which the three significant figures like

asked for in the question would be 3.62

so the length of our missing side is

3.62 centimeters

now one thing to point out which can

sometimes be a bit confusing is that

most examined questions will label the

corners for you like they have in this

one and when they refer to a line they

use the corners it's connected to

so when they ask us to find the length

of x z

they are asking us to find this unknown

side between x and z

so we could put a question mark on this

side because this is the one that we're

trying to find

because we're going to have to use

pythagoras theorem though we want to

label our sides a b and c

using the rules that we've already been

talking about

so from this point we can pretty much

ignore the x y and z

so we write out our equation a squared

plus b squared equals c squared and plug

in the values

so 5.6 squared plus 10.5 squared equals

c squared

which if you put it into the calculator

simplifies to 141.61

equals c squared

and if we square root both sides we find

that c is equal to 11.9

and so the length of xz is 11.9

centimeters

let's try one more in this style

so in this question we're trying to find

the length of ac which is this one

so we can give this line a question mark

and from this point onwards we can

ignore the letters that they've given

the corners

so we need to label our sides a b and c

like we would for any pythagoras

question

and write out the equation

a squared plus b squared equals c

squared

so that would mean that 8 squared plus

11 squared equals c squared

which simplifies to 64 plus 121 equals c

squared

or 185 equals c squared

and if we square root both sides that

gives us 13.6 equals c

and so we can write our answer as 13.6

centimeters

anyway that's everything for this video

so hope you enjoyed it please do give us

a like and subscribe and remember to

check out our website by clicking on the

link in the top right corner of this

screen

and we'll see you again soon