Table of Contents
In this lesson, we will learn about the Pythagorean Theorem. or the Pythagorean theorem as it is sometimes called. And you might be wondering, “What is the theorem?” and “Who in this world is Pythagoras?” Now, in mathematics, theorems are only statements that have been proven to be true. from other things that are known or accepted to be true. And Pythagoras, well, he’s a very smart man. who lived a long time in ancient Greece. and he proved the theorem.
Well historians are not entirely sure it was actually Pythagoras theorem who proved it. It could be one of the students or followers. but he usually gets praise for it. However, the main thing you need to know is that. Pythagoras’s theorem illustrates important geometric relationships. between the three sides of a right triangle.
We will learn what the relationship is in just one minute. but first there are a few things you need to know. before you can really understand the Pythagorean Theorem or use it to solve problems.
First of all, to understand the Pythagorean Theorem you need to know about angles and triangles. and you also need to know a little about exponents and square roots. So, if the topic is new to you, be sure to watch our videos about them first. And secondly, although the Pythagorean Theorem is about geometry.
You need to know some basic algebra to really use it. Specifically, you have to know about variables. and how to solve basic algebraic equations involving exponents. We discussed many of these topics in the first five videos of our Basic Algebra series.
Read also : Algebra Basic
Okay, now you have all the background information discussed. let’s see what the Pythagorean Theorem says. The theorem can be expressed in several different ways. but what we like the most is like this: For right triangles with feet ‘a’ and ‘b’ and sloping sides ‘c’.
‘a squared’ plus ‘b squared’ is equal to ‘c squared’. As you can see from this definition, the Pythagorean Theorem does not apply to ALL triangles. ONLY applies to the RIGHT triangle. As you know, a right triangle always includes a right angle.
which is usually marked with a ‘right-angle symbol’ of a square to help you identify it. And you need to know which angle is the right angle because. this helps you identify the important side of the triangle called the hypotenuse.
The hypotenuse is the longest side of a right triangle. and always the ‘opposite’ side from the right angle. In other words, the side that doesn’t touch (or helps shape) the right angle itself. To use the Pythagorean Theorem, you must be able to identify the hypotenuse.
because that’s what is the variable ‘c’ for the theorem. ‘c’ is the length of the hypotenuse. The other two sides of the triangle. (which touches or forms a right angle) is called “his feet”.
Our Pythagorean Theorem definition uses the variable names ‘a’ and ‘b’ to represent their length. Oh and which foot is called ‘a’ and which foot is called ‘b’.
as long as you keep track of which after you make your initial choice. Okay, now we know the various parts of the Pythagorean Theorem. let’s think about what relations or equations (‘a squared’ plus ‘b squared’ equal to ‘c squared’) really tell us.
This tells us that if we take the lengths of both legs (sides ‘a’ and ‘b’) and their ‘square’. which means multiplying it yourself. (‘a squared’ is ‘a’ times ‘a’ and ‘b squared’ is ‘b’ times ‘b’). and then if we add the two ‘sums of squares’ together.
they will EQUAL the amount that you will get if you ‘square’ the hypotenuse. (which will be ‘c squared’ or c times c). That might sound a little confusing at first. So let’s look at a specific example of a right triangle. it will help the Pythagorean theorem make more sense. This right triangle is called “3, 4, 5 triangles”.
because the sides have relative lengths 3, 4, and 5. And with “relative length”, I mean that the unit of length doesn’t really matter. Sides can be expressed in ANY unit (inches, meters, miles, whatever).
So a triangle can be of various sizes along its length. will have proportions of 3, 4, and 5 relative to each other. Starting with the side that is 3 units long, which we call “side a”. what do we get if we square that side ?
Now in arithmetic, quadratic 3 means multiplying 3 times 3 which is equal to 9. And the geometric equation of squaring something actually produces a square shape. As you can see, this box contains 9 unit boxes.
So this red area represents the value of ‘a’ which is squared in the Pythagorean Theorem. Next let’s look at the side which is 4 units in length, which we call “side b”. Squaring 4 means multiplying 4 times 4, which is 16.
Again, the geometric equivalent is a literal square. i.e. 4 units on each side and cover a total area of 16 units. So this blue area represents ‘b squared’ in the Pythagorean Theorem. And finally, let’s deal with the hypotenuse, or “c side”, which is the longest side.
5 units in length. Squaring 5 means multiplying 5 times 5, which is 25. And the geometric equation is 5 x 5 square which has an area of 25 units. So this green area represents ‘c squared’ in the Pythagorean Theorem.
Now you can see how the arithmetic part of the Pythagorean Theorem. related to the geometric parts of this right triangle. let’s check to see if the Pythagorean Theorem is really true (at least in this particular case).
On the arithmetic side, if you add up the sum of ‘squared’ and ‘b squared’. they are really the same as ‘c squared’ because 9 + 16 = 25. And by slightly rearranging our unit box. You can see that the area of the box is formed by two legs. really the same as the area of the square formed by the hypotenuse.
Wow, those ancient Greek men are really smart! Okay, but I know what you think. “It’s okay, but why should I care about the Pythagorean Theorem? What are the benefits? “. As always, that’s a good question. And the answer is, like many things in mathematics.
The Pythagorean theorem is a useful tool. that can help you use what you do NOT know to find out what you do NOT know. Specifically, if you have a right triangle but you only know how long the two sides are.
The Pythagorean theorem tells you how to find out the length of an unknown third side. For example, imagine that you have a right triangle. 2 cm long on this side and 3 cm long on this side. but we don’t know how long the tilt is.
No problem!. The Pythagorean theorem tells us the relationship between the three sides of ANY right triangle. so we can find out. We know that ‘a squared’ plus ‘b squared’ is equal to ‘c squared’. so let’s plug what we know into the equation. and then solve it for what we don’t know.
Again, it doesn’t matter which leg is called ‘a’ or ‘b’. so let’s label it like this. and then substitute ‘2’ for ‘a’ and ‘3’ for ‘b’ in the Pythagorean Theorem equation. That gives us an algebraic equation that has only one unknown, ‘c’. If we solve this equation for ‘c’ in other words.
if we rearrange the equation so that ‘c’ by itself on one side of the sign is equal to. then we will know exactly what ‘c’ is. We will know the length of the side of the triangle. First, we need to simplify the left side of the equation because it contains known numbers. And according to the order of operations, we need to simplify the exponent first.
‘2 squared’ is 4 and ‘3 squared’ is 9. Then, we add the result (4 + 9 = 13). and we have equation 13 = ‘c squared’. which is equal to ‘c squared’ = 13. Then, to get ‘c’ by itself. we need to do the opposite of what was done to him.
Because it is squared, the reverse operation is square root. so we need to take the square root from both sides. Taking the square root of ‘c squared’ only gives us the ‘c’ we want on the side of this equation.
but that gives us a little problem on the other side. because it’s not easy to know what the square root of 13 is. This is not a perfect square so it will be decimal. and maybe irrational numbers.
But that’s okay because it leaves our answer as ‘square root of 13’. Of course, you can use a calculator to get decimal values if you really need them. but in mathematics it is very common to leave square roots alone unless they are easy to simplify.
So the sides of this right triangle are, 2 cm, 3 cm and the ‘square root of 13’ cm. Let’s try another example. For this right triangle, we know the length of the hypotenuse (6 m). and one foot (4 m). but the length of the other leg is unknown.
So let’s use the Pythagorean Theorem to find the unknown length. As usual, we call the hypotenuse “c side”. And let’s call our feet know “side a” and our feet don’t know “side b”. Then we can replace the known values into the Pythagorean Theorem.
and solve for unknown values. Replacing ‘c’ with 6 and ‘a’ with 4 gives us the equation. ‘4 squared’ plus ‘b squared’ is equal to ‘6 squared’. we need to simplify and solve for ‘b’. First let’s simplify the exponent. ‘4 squared’ is 16, and ‘6 squared’ is 36.
Now we need to isolate ‘b squared’. and we can do that by subtracting 16 from both sides of the equation. On this side, (+16) and (-16) leave us only with ‘b squared’. And on the other hand we have 36 minus 16 which is 20.
We can now solve this simplified equation for ‘b’. by taking the square root from both sides which gives us. ‘b’ is the same as ‘square root of 20’.
Again, it’s okay to leave your answer as a square root like this. And some of you might know that ‘square root of 20’ can be simplified. to ‘2 times the square root of 5’. We will not worry about simplifying the root in this video. but if you know how to do it, great! If you don’t know, just leave the answer as ‘square root of 20’ meters.
Here’s another interesting one. What if you have a ‘unit square’ cut in half along the diagonal. Each side of the square is 1 unit. but how far is the distance from one corner of the square to another along the diagonal ?
Well, because the diagonal divides the square into two right triangles. we can use the Pythagorean Theorem to tell us the unknown distance. We label the right triangle feet ‘a’ and ‘b’ and the hypotenuse of ‘c’.
And because we know that ‘a’ and ‘b’ are equal to 1, we can attach those values. into the equation of the Pythagorean Theorem that gives us us. ‘1 squared’ plus ‘1 squared’ is equal to ‘c squared’. Now we are done with ‘c’. ‘1 squared’ is only 1. so the left side of this equation simplifies to 1 plus 1 which is only 2.
That means ‘c squared’ is equal to 2. and if we take the square root from both sides. we get ‘c’ equal to ‘square root of 2’. So that’s how far it crosses the diagonal of the unit square. Okay, so that’s how you use the Pythagorean Theorem.
to find the unknown side length of a right triangle, which is the most common use. But there are other ways you can use the Pythagorean Theorem that I want to mention. You can also use the Pythagorean Theorem to TEST triangle.
to see if it’s really the RIGHT triangle. Yes, know if you are not sure. For example, what if someone shows you this triangle and asks. “Is this a right triangle?” Yeah, that looks a lot like a right triangle. but does not have a right-angle symbol. and it will be difficult to know whether this angle is exactly 90 degrees just by looking at it.
Maybe close to 90, like 89.5 degrees. Don’t worry, the Pythagorean Theorem can tell us for sure. if we know the length of all three sides of the triangle. If we know the length (a, b and c). then we can include them in the Pythagorean Theorem equation to see if that is true.
In this special case. because the two shorter sides are 3 cm each and the longest side is 4 cm. we can attach the values to ‘a’, ‘b’ and ‘c’ and simplify to see what we get. ‘3 squared’ is 9, so on this side of the equation we get 9 plus 9 which is 18. And on the other side we have ‘4 squared’, which is 16. Uh oh that’s not right!
Our equation is simplified to 18 equals 16. What is clear is NOT a true statement. That means that all three sides of this triangle do not function in the Pythagorean Theorem. they do not match the relation ‘a squared’ plus ‘b squared’ equal to ‘c squared’.
And because the Pythagorean Theorem tells us that ALL right triangles fit that relationship. this triangle can’t be the RIGHT triangle. Good. so now you know what the Pythagorean Theorem is and you know how to use it.
You can use it to find the missing side of the right triangle. and you can also use it to test a triangle to see if it qualifies as a right triangle. But as you see. many other mathematical skills are needed to be able to use the Pythagorean Theorem effectively.
so you might need to brush up on some of those skills before you are ready to try using them yourself. And remember, you can’t master math just by watching videos about it. You really need to practice solving real math problems.
Pythagoras Conference Global
Evidence Pythagoras signs is frequently called the definition of Pythagoras theorems (Pitagoras). Because as the Elementary school when learning math, we don’t overlook to examine phytagoras this sentence, Pythagoras has to have already been no stranger inside our ears. The phytagoras formula is just actually a formula utilized by way of a scientist.
The significance of Pythagoras or even pythagoras theorem reads:
The left-hand or the side at the elbow’s triangle is corresponding to the flip hand squared.
The triangle above would be just an angled triangle, which includes one side vertical (BC), 1 side (AB), and also yet one side tilts (AC).
Evidence pythagoras conference global and also the Pythagoras formula functions to locate 1 side with both sides that are known.
Pythagoras theorem calculator
- Pythagoras formulation in origin shape
- In the event the mirror C
- The vertical and forthcoming sides are b and A
The consequent Pitagoras formulation:
b² = a² + c²
Then to calculate the upright side and the upcoming side apply formula:
a² = b² – c²
c² = b² – a²
Pythagoras formula in root form
- If the mirror side C
- The upright and upcoming sides are A and b
Significant NOTE: The Pythagoras formula, just valid on the elbows only (Pythagoras formula in square roots).
From the signs of this Pythagorean theorem, There’s a blueprint of amounts that Will Need to be recalled to solve the Challenge of Pythagoras is likely to likely probably be simpler and faster to operate on this, the routine is:
- 3 – 4 – 5
- 5 – 12 – 13
- 6 – 8 – 10
- 7 – 24 – 25
- 8 – 15 – 17
- 9 – 12 – 15
- 10 – 24 – 26
- 12 – 16 – 20
- 14 – 48 – 50
- 15 – 20 – 25
- 15 – 36 – 39
- 16 – 30 – 34
To know more information about the signs of Phytagoras, subsequently think about the next instance
Cases of pythagoras theorem and conversation
Example 1
A concentric elbow comes with a vertical side (a b ), which is 1-5 cm long, and also alongside, it is (BC ) 8 cm, what’s cm kah side of this Mirror (AC)?
Understood:
A B = 1-5
B C = 8
Said: AC span…???
⇒ First Way
AC² = AB² + BC²
AC² = 15² + 8²
AC² = 225 + 64
AC² = 289
AC = √289
AC = 17
⇒ Second way
AC = √ AB² + BC²
AC = √ 15² + 8²
AC = √ 255 + 64
AC = √ 289
AC = 17
So, AC length is 17 cm
Example 2
Just how long is the amount of the borders of an elbow-elbow if it’s called this period of the mirror 1 3 cm and also the data side 5 cm?
Its conclusion:
As an instance: c Tilt side-by-side Flat side, an upright side
Not Known: c = 1 3 cm, B = 5 cm
Said: a …????
⇒ First Way
a² = c² – b²
a² = 13² – 5²
a² = 169 – 25
a² = 144
a = √ 144
a = 12
⇒ Second Way
a = √ c² – b²
a = √ 13² – 5²
a = √ 169 – 25
a = √ 144
a = 12
So, the length of the upright triangle is 12 cm
Example 3
There’s a triangle of ABC, elbows in B. In case the span is a b = 16 cm and BC = 30, then what’s the amount of the side of this triangle (AC)?
Understood:
A B = 16
B C = 30
Said: AC =…?
AC = √ AB² + BC²
AC = √ 16² + 30²
AC = √ 256 + 900
AC = √ 1156
AC = 34
So, AC length = 34 cm
Last Exam
Notice the picture below, learn the ABC elbow has a vertical side significance of 6 cm and a bottom side of 8 cm, then calculate just how long would be your medial side of this mirror?
Understood:
A B = 8 cm
B C = 6 cm
Length: Period of AC (angled side of this elbow-elbow above)…?
AC² = AB² + BC²
AC² = 8² + 6²
AC² = 64 + 36
AC² = 100
AC = √100
AC = 10
These are a few situations of pythagoras calculator and also their talk and responses. To understand you, please perform a little exercise about learning Phytagoras below.
Try Out
1. There’s just actually a triangle PQR X Y Z called the sides, which can be Y, X, and Z. In the subsequent announcement, the stark reality is…?
- A. if y² = x² + z² , < X = 90º
- B. if z² = y² – x² , < Z = 90º
- C. if z² = x² – y² , < Y = 90º
- D. if x² = y² + z² , < X = 90º
2. Not known triangle PQR has elbow at Q, where P Q = 8 cm, PR = 17 cm. Therefore, along this QR will be…?
- A. 9 cm
- B. 15 cm
- C. 25 cm
- D. 68 cm
3. There’s a squat triangle, its hypotensive 4 √3 cm along with a single facet of this elbows is 2 √2 cm. The length of time can be just the rear side of the elbow… Cm
A. 2 √10
B. 3 √5
C. 8 √2
D. 3 √3
4. The period of this Hepotenusa Tri-angle in which the leg is the span that is elbow and also 16 cm is x 5. Figure out the price of X …. Cm
A. 4 √2
C. 8 √2
D. 8 √3
Therefore the reason for this sign pythagoras conference global, ideally helpful and certainly will assist in learning math, which frequently makes us all around. Once the initial when we study, then all of the problematic things will probably be more comfortable.
The gist of this Pythagoras formula could be that your angled side adds up to the vertical side on both sides and a horizontal border (however, remember to get acquainted).
