The Pythagorean Theorem

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What Does the Pythagorean Theorem Say?

The Pythagorean Theorem can be formulated as follows: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

In the right triangle shown in the image below, we use the first letters of the alphabet to indicate its sides:

a a and b b are the legs.

cc is the hypotenuse.

Using these, we can express the Pythagorean theorem in an algebraic form as follows:

c2=a2+b2 c²=a²+b²

How to solve the Pythagorean theorem

We can express the Pythagorean Theorem in a geometric form in the following way, showing that the area of the square (c c ) (square of the hypotenuse) is the sum of the areas of the squares (a a ) and (b b ) (squares of the legs).

geometrical form of the Pythagorean Theorem

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The Pythagorean Theorem: Explanation and Examples

The Pythagorean Theorem is one of the most famous theorems in mathematics and one of the most feared topics among students. It is no coincidence that it is among the most common mathematical theorems and one that is likely to be encountered outside of your studies and exams.

This theorem is attributed to Pythagoras of Samos. Born in 570 BC, he was a Greek scholar to whom we also owe the word 'philosopher'.

The Pythagorean Theorem establishes the relationship between the three sides of a right triangle.

In this article, we will explain in a simple and practical way what the Pythagorean Theorem is and give you some examples. Let's jump in!

To begin with, it is essential to clarify some important points:

  • A triangle is a polygon with three sides.
  • The angles of any triangle add up to 180o 180^o degrees.
  • A right triangle is a triangle in which one of the angles equals 90o 90^o degrees.
  • A right angle is an angle of 90o 90^o degrees.
  • In every right triangle, the sides adjacent to the right angle are called "legs". The legs are the sides that form the right angle.
  • The longest side of a right trianglethe one opposite the right angleis called the "hypotenuse".

What is a Theorem?

We can use the Pythagorean Theorem to understand what a theorem is.

A theorem is a demonstrable statement that links two propositions. We start from a first proposition that we call a hypothesis to assert a second proposition that we call a thesis.

The statement of a theorem affirms that if the hypothesis is true, then the thesis is also true.

The proof of a theorem is the most difficult part and is usually left to mathematicians. The important thing is that once a theorem is proved, we can confidently use the statement of the theorem as a permanent truth.

Returning to the Pythagorean Theorem, let us highlight which is the hypothesis and which is the thesis. To do this, we reformulate the statement of the theorem using the expressions if and then as follows:

The Pythagorean Theorem states that:

If:

A triangle is right angled (a triangle containing an angle of 90o 90^o ) (hypothesis).

Then:

The square of the longest side of the triangle is the sum of the squares of the other two sides (thesis).


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Reciprocal of the Pythagorean Theorem

In the Pythagorean Theorem, the reciprocal of the theorem is also true, For example:

If:

  1. The square of the longest side of a triangle is the sum of the squares of the other two sides (hypothesis).

Then:

2. The triangle is right angled (the value of one of the angles of the triangle is 90o 90^o ) (thesis).


What is the Pythagorean Theorem Used For?

The Pythagorean Theorem is arguably a cornerstone of Cartesian geometry and has therefore become an important driving force in the development of the sciences as we know them today.

The importance of this theorem comes from the significance of the right triangle, which is a triangle that links a horizontal line with a vertical line (the legs of the triangle). The horizontal and vertical lines always form an angle of 90o 90^o .

The Pythagorean Theorem is applicable in all areas of science as they all share a mathematical basis.


Do you know what the answer is?

Exercises for Using the Pythagorean Theorem

Below we present several problems with the Pythagorean theorem:


Exercise 1

Find the value of X X in the following triangle:

The Pythagorean Theorem Exercise 1

Solution:

The picture shows a triangle. We know the length of two of its sides, but we want to work out the value of the third side.

We also know that the triangle shown is a right triangle because a small box indicates a right angle.

The Pythagorean theorem says that in a triangle the following is true:

c2=a2+b2 c²=a²+b²

In our right triangle:

a=3 a= 3

b=4 b=4

c=X c=X

By substituting the values of our triangle in to the algebraic expression of the Pythagorean Theorem, we get the following equation:

X2=32+42 X²=3²+4²

X2=9+16 X²=9+16

X2=25 X²=25

If we now remove the square root from both sides of the equation, we can work out X X and obtain the value we are looking for:

X=25 X=\sqrt{25}

X=5 X=5


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Exercise 2

Calculate the length of X X in the following right triangle:

The Pythagorean Theorem exercise 2 new

The image shows a right triangle with two known side lengths and one unknown side length. We want to work out the length of the third side—in this case, one of the legs.

The Pythagorean theorem states that the following is true in a right triangle:

c2=a2+b2 c²=a²+b²

In our right triangle:

a=3 a= 3

b=X b=X

c=5 c=5

By substituting the values of our triangle in to the algebraic expression of the Pythagorean Theorem, we obtain the following equation:

52=32+X2 5²=3²+X²

25=9+X2 25=9+X²

To work out the value of X X , we must begin by subtracting 9 9 from each side of the equation.

259=X2 25-9=X²

16=X2 16=X²

Next, by removing the square root from both sides of the equation, we get the value of X X .

16=X \sqrt{16}=X

X=4 X=4

Therefore, the answer is: 4 4 .


Exercise 3

What is the value of X X in the triangle shown in the following image?

Pythagorean Theorem exercise 3

Solution:

We are dealing with a right triangle which is also isosceles since two of its sides have the same length.

In this case, we know the length of the hypotenuse of the right triangle and we want to work out the length of each leg, knowing that both legs have the same value.

The Pythagorean theorem says that the following is true for a right triangle:

c2=a2+b2 c²=a²+b²

In our right triangle:

a=X a= X

b=X b=X

c=10 c=10

By substituting the values of our triangle in to the algebraic expression of the Pythagorean Theorem, we get the following equation:

102=X2+X2 10²=X²+X²

100=2X2 100=2X²

To work out X, we must begin by dividing each side of the equation by 2 2 .

1002=2X22 \frac{100}{2}=\frac{2X²}{2}

50=X2 50=X²

Finally, we need to remove the square root from both sides of the equation to obtain the value of X X .

50=X \sqrt{50}=X

X=7.07 X=7.07


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What is the Most Common Use of the Pythagorean Theorem?

The Pythagorean Theorem is mainly used in exercises related to right triangles in which the length of both legs is given to find the length of the hypotenuse.


The Inverse of the Pythagorean Theorem

There is also the inverse theorem by which we can prove that a given triangle is right-angled: a triangle in which the sum of both legs squared is equal to the hypotenuse squared is a right-angled triangle.


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Pythagorean Theory Exercises

Exercise 1

Given the triangle:

Given the triangle question 180

Task:

What is the value of X X ?

Solution:

Don't fall into the trap!

Remember, in the Pythagorean Theorem only a right triangle can be used. This triangle is an obtuse triangle, so the theorem does not apply to it.

Answer:

It is not possible to calculate the answer using the Pythagorean theorem.


Exercise 2

Assignment:

Given the triangle ABC \triangle ABC , find the length of BC BC .

Exercise 2 Given triangle ABC

Solution:

Apply the formula.

Given the triangle ABC \triangle ABC in the drawing, write the Pythagorean Theorem for the right triangle: ABC \triangle ABC

AB2+BC2=AC2 AB²+BC²=AC²

We substitute in the known lengths:

52+BC2=132 5²+BC²=13²

25+BC2=169 25+BC²=169

BC2=16925=144 BC²=169-25=144 , \sqrt{}

BC=12 BC=12

Answer:

12 12 cm.


Do you know what the answer is?

Exercise 3

Task:

Given the triangles in the drawing, what is the length of the side DB DB ?

Exercise 3 Given the triangles in the drawing

Solution:

We start with the triangle ABC \triangle ABC , writing down the Pythagorean Theorem as an equation:

BC2+BA2=AC2 BC²+BA²=AC²

We then substitute in the given sides from the diagram:

BC2+62=(211)2 BC²+6²=(2\sqrt{11})²

BC2+36=22112=4×11=44 BC²+36=2²\sqrt{11}²=4\times11=44

BC2+36=44 BC²+36=44 /11 -11

BC2=4436=8 BC²=44-36=8 / \sqrt{}

BC=8 BC=\sqrt{8}

Now, we write down the equation for the triangle BCD \triangle BCD :

DC2+DB2=BC2 DC²+DB²=BC²

Finally, we insert the values of the sides CD CD, BC BC:

22+DB2=(8)2 2²+DB²=(\sqrt{8})²

4+DB2=8 4+DB²=8 /4 -4

DB2=4 DB²=4 / \sqrt{}

DB=2 DB=2

Answer:

DB=2 DB=2


Exercise 4

Task:

Exercise 4 Given the drawing of a triangle

The triangle ABC \triangle ABC is given.

The ratio of BCBC to the hypotenuse ACAC is 1:4 1:4:

AB=315 AB=3\sqrt{15}

Solution:

Write the Pythagorean Theorem as an equation for the triangle: ABC ABC :

AB2+BC2=AC2 AB²+BC²=AC²

For the hypotenuse ACAC ratio with BC BC we can write:

BCAC=14 \frac{BC}{AC}=\frac{1}{4}

Therefore:

BC=14AC BC=\frac{1}{4}AC

Substituting our values in to the formula:

(315)2+(14)2AC2=AC2 (3\sqrt{15})²+(\frac{1}{4})²AC²=AC²

32152+116)2AC2=AC2 3²\sqrt{15}²+\frac{1}{16})²AC²=AC²

We perform the operations in the formula:

116AC2 -\frac{1}{16}AC²

9×15=AC216AC2=1516AC2 9\times15=AC²-\frac{1}{6}AC²=\frac{15}{16}AC² / We multiply by 1615 \frac{16}{15}

9×15×1615=AC2 9\times15\times\frac{16}{15}=AC² / We apply the root.

\sqrt{}

9×16=9×16=3×4=12=AC \sqrt{9\times16}=\sqrt{9}\times\sqrt{16}=3\times4=12=AC

Answer:

AC=12cm AC=12cm


Check your understanding

Exercise 5

Yoel goes up a bicycle ramp to a maximum height of 3 3 meters above the ground.

All other measurements are shown in the figure:

bicycle ramp at a maximum height of 3 meters above the ground

On the way up Yoel, travels at a speed of 2 2 meters per second.

On the way down, he travels at a spreed of 6 6 meters per second.

Task:

How long will it take Yoel to go over the ramp?

Solution:

Ascent:

Find the hypotenuse of the right triangle that has legs measuring 3 3 and 7 7 meters long.

32+72=(hypotenuse)2 3²+7²=(hypotenuse)²

9+49=(hypotenuse)2 9+49=(hypotenuse)²

hypotenuse=58=7.62 hypotenuse=\sqrt{58}=7.62

time=distancespeed=7.622=3.81  time=\frac{distance}{speed}=\frac{7.62}{2}=3.81\text{ }

Descent:

Find the hypotenuse of the right triangle the legs of which are 3 3 and 4 4 meters long.

(hypotenuse)2=32+42 (hypotenuse)²=3²+4²

\sqrt{} / (hypotenuse)2=9+16 (hypotenuse)²=9+16

5=25 5=\sqrt{25}

Hypotenuse: 5 5 meters.

time=distancespeed=56=0.83 \text{time}=\frac{distance}{speed}=\frac{5}{6}=0.83

time=3.81+0.83=4.64 \text{time}=3.81+0.83=4.64

Answer:

Yoel will cross the ramp in 4.64 4.64 seconds.


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