GCSE Maths - Pythagoras' Theorem And How To Use It #120
Summary
TLDRThis 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
- đ To use Pythagoras' theorem, the triangle must be a right-angled triangle with one 90-degree angle.
- đ 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.
- â 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.
- đ It's important to memorize the equation as it's frequently used in mathematics.
- đ The order of a and b doesn't matter, but c must be the hypotenuse opposite the right angle.
- 𧟠Substitute the known side lengths into the equation and solve for the unknown side.
- đ When solving, simplify the equation and use mathematical operations to isolate the unknown side.
- đ 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 = â25 = 5 units.
- đ In exam questions, the sides are often labeled with letters, and you should label them as a, b, and c for clarity.
- đą 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
đ 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 aÂČ + bÂČ = cÂČ 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.
đą 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.
đ 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.
Mindmap
Keywords
đĄPythagorean Theorem
đĄRight-Angle Triangle
đĄHypotenuse
đĄSides a and b
đĄEquation
đĄCalculation
đĄSquare Root
đĄUnknown Side
đĄLabeling Sides
đĄSignificant Figures
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
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