Pythagorean Theorem - Examples, Exercises and Solutions

We use the Pythagorean theorem

$AC^2+AB^2=BC^2$

We insert the known data:

$3^2+4^2=BC^2$

$9+16=BC^2$

$25=BC^2$

We extract the root:

$\sqrt{25}=BC$

$5=BC$

To find side AB, we will need to use the Pythagorean theorem.

The Pythagorean theorem allows us to find the third side of a right triangle, if we have the other two sides.

You can read all about the theorem here.

Pythagorean theorem:

$A^2+B^2=C^2$

That is, one side squared plus the second side squared equals the third side squared.

We replace the existing data:

$3^2+2^2=AB^2$

$9+4=AB^2$

$13=AB^2$

We find the root:

$\sqrt{13}=AB$

To solve the exercise, we have to use the Pythagorean theorem:

A²+B²=C²

We replace the data we have:

3²+4²=C²

9+16=C²

25=C²

5=C

To find the length of the hypotenuse BC in a right-angled triangle where AB and AC are the other two sides, we use the Pythagorean theorem: $c^2 = a^2 + b^2$.

Here, $a = 6 \text{ cm}$ and $b = 8 \text{ cm}$.

Plugging the values into the Pythagorean theorem, we get:

$c^2 = 6^2 + 8^2$.

Calculating further:

$c^2 = 36 + 64$

$c^2 = 100$.

Taking the square root of both sides gives:

$c = 10 \text{ cm}$.

We use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

It is important to remember: the Pythagorean theorem is only valid for right-angled triangles.

This triangle does not have a right angle, and therefore, the missing side cannot be calculated in this way.

To solve the exercise, it is necessary to know the Pythagorean Theorem:

A²+B²=C²

We replace the known data:

2²+B²=7²

4+B²=49

We input into the formula:

B²=49-4

B²=45

We find the root

B=√45

This is the solution. However, we can simplify the root a bit more.

First, let's break it down into prime numbers:

B=√(9*5)

We use the property of roots in multiplication:

B=√9*√5

B=3√5

This is the solution!

To answer this question, we must know the Pythagorean Theorem

The theorem allows us to calculate the sides of a right triangle.

We identify the sides:

ab = a = 5
bc = b = ?

ac = c = 13

We replace the data in the exercise:

5²+?² = 13²

We swap the sections

?²=13²-5²

?²=169-25

?²=144

?=12

We use the Pythagorean theorem as shown below:

$AC^2+BC^2=AB^2$

Since the triangles are isosceles, the theorem can be written as follows:

$AC^2+AC^2=AB^2$

We then insert the known data:

$2AC^2=(8\sqrt{2})^2=64\times2$

Finally we reduce the 2 and extract the root:

$AC=\sqrt{64}=8$

$BC=AC=8$

Let's focus on triangle BCD in order to find side BC.

We'll use the Pythagorean theorem using our values:

$BC^2+DC^2=BD^2$

$BC^2+24^2=25^2$

$BC^2=625-576=49$

Let's now remove the square root:

$BC=7$

Since each pair of opposite sides are equal to each other in a rectangle, we can state that:

$DC=AB=24$

$BC=AD=7$

Now we can calculate the perimeter of the rectangle by adding all sides together:

$24+7+24+7=14+48=62$

Let's focus on triangle BCD in order to find side DC.

We'll use the Pythagorean theorem and input the known data:

$BC^2+DC^2=BD^2$

$6^2+DC^2=10^2$

$DC^2=100-36=64$

Let's now remove the square root:

$DC=8$

Since in a rectangle each pair of opposite sides are equal to each other, we know that:

$DC=AB=8$

$BC=AD=6$

Now we can calculate the perimeter of the rectangle by adding all sides together:

$8+6+8+6=16+12=28$

To solve this problem, we'll follow these steps:

• Step 1: Use the Pythagorean Theorem to calculate the length of $AD$.
• Step 2: Calculate the area of triangle $\triangle BCD$.

Step 1: Given $AB = 12$ and $AC = 13$, we use the Pythagorean Theorem to find $AD$.

Step 2: Knowing the sides $AD = 5$ (height of the rectangle) and $AB = 12$ (base of the rectangle), triangle $\triangle BCD$ will have the base $BC = 12$ and the height $BD = 5$.

The area of triangle $\triangle BCD$ is:

Therefore, the area of triangle $\triangle BCD$ is 30.

In this question, we will have to use the Pythagorean theorem twice.

A²+B²=C²

Let's start by finding side CB:

6²+CB²=(2√11)²

36+CB²=4*11

CB²=44-36

CB²=8

CB=√8

We will use the exact same way to find side DB:

2²+DB²=(√8)²

4+CB²=8

CB²=8-4

CB²=4

CB=√4=2

To find the length of the hypotenuse of the triangle, let's proceed with the following steps:

• Step 1: Use the area formula to find the base.
• Step 2: Apply the Pythagorean Theorem to find the hypotenuse.

Step 1:
The formula for the area of a right-angled triangle is:

Given that the area is 30 cm², and one leg (the height) is 5 cm, we can find the base:

Solve for $\text{base}$:

Multiply both sides by 2:

Divide by 5:

Step 2:
With base and height (legs of the triangle) known, apply the Pythagorean theorem to find the hypotenuse, $c$:

Calculate:

Take the square root to find $c$:

Therefore, the length of the hypotenuse of the triangle is 13 cm.

This problem involves determining the lengths of the legs of a triangle whose perimeter is 12 cm, given that one side is 5 cm. To solve, consider the apparent context that implies a right triangle.

First, let's denote the three sides of the triangle as $a$, $b$, and $c$, where $c = 5$ cm.

Considering the perimeter formula:

$a + b + c = 12$

Since $c$ is 5 cm, the equation becomes:

$a + b + 5 = 12$

Solving for $a + b$:

$a + b = 7$

Assuming it is a right triangle with the side length of 5 cm as the hypotenuse:

$c^2 = a^2 + b^2$

Where $c = 5$, the equation is:

$5^2 = a^2 + b^2$
$25 = a^2 + b^2$

We need the integers $a$ and $b$ that satisfy both $a + b = 7$ and $a^2 + b^2 = 25$.

To trial integer pairs from $a + b = 7$:

- If $a = 3$, then $b = 4$.

Check $a = 3$ and $b = 4$ in the Pythagorean condition:

$3^2 + 4^2 = 9 + 16 = 25$

Hence, the pair satisfies both conditions.

Therefore, the lengths of the legs are $3 \, \text{cm}$ and $4 \, \text{cm}$.