Pythagorean Theorem
- Who was Pythagoras?
- Pythagorean Theorem
- Pythagorean Formula
- Pythagorean Triples
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Who Was Pythagoras?
Pythagoras was a Greek mathematician who discovered the solution to find the length of the missing side of a right-angle triangle when the measure of the other two sides is known. Since he discovered the theorem, it is named after him and called Pythagoras theorem. It shows the connection between all the sides of a right-angle triangle.
Pythagoras Theorem (a.k.a. Pythagorean Theorem)
It is important to note that Pythagoras theorem is only applicable to right-angled triangles. According to the theorem, the square of the hypotenuse of the right triangle is equal to the sum of the squares of the other two sides. The hypotenuse is the side with the maximum length or opposite to the right angle.
Suppose, `ABC` is a right triangle at `B` where `AC` is the hypotenuse and `AB` and `BC` are the legs of the triangle, in other words, `AC` is the side opposite to the right angle, `AB` is the perpendicular side (altitude of the triangle), and `BC` is the base of the triangle.
Note: `AC = \text{Hypotenuse}`
`AB = \text{Height}`
`BC = \text{Base}`
Pythagorean Formula
When we are given the measures of any two sides of a right-angle triangle and we are supposed to find the measure of the missing side, we use the Pythagorean formula. To calculate the measurement of any side we can use this formula.
According to the Pythagorean formula,
`AC^2 = AB^2 + BC^2`
or,
`\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2`
Look at the given figure, it represents a right-angle triangle with three different lengths of square.
In this figure,
Area of the pink box `=` Area of the green box `+` Area of the pitch box
Area of `Y` square `=` Area of `X` square `+` Area of `Z` square.
This emphasizes the Pythagoras Theorem.
`Y^2 = X^2 + Z^2`
Or, `AC^2 = AB^2+ BC^2`
We can use the Pythagorean formula to conclude if the `3` given sides of a triangle can form a right-angle triangle or not.
Pythagorean Triples
Pythagorean theorem and Pythagorean triples go hand in hand. In simple words , Pythagorean triples are the integer solutions to the Pythagorean theorem, containing positive integers.
We represent the Pythagorean triples as `(a, b, c)`. Here, `“c”` is the “hypotenuse” or the longest side of the triangle, and `“a”` and `“b”` are the other two sides of the right-angled triangle. The three sides, `“a”`, `“b”`, and `“c”` of any right angle triangle satisfy `c^2 = a^2 + b^2` .
There is an endless list of Pythagorean triples. Here are a few common Pythagorean triples:
`3`-`4`-`5`
`5`-`12`-`13`
`6`-`8`-`10`
`9`-`12`-`15`
`12`-`16`-`20`
Let us verify if `6`,`8` and `10` satisfy `c^2 = a^2 + b^2` .
`6^2+8^2=36 + 64 = 100 =10^2`
This proves that `6`,`8` and `10` are Pythagorean triples. Meaning it is possible to draw a triangle whose hypotenuse is `10` units and the other two sides are `6` units and `8` units.
New triples can be formed by multiplying all the numbers of an original Pythagorean triple by another positive integer. For example, we can form the following Pythagorean triples from `3`-`4`-`5`:
`6`-`8`-`10`
`9`-`12`-`15`
`12`-`16`-`20` … and so on.
We can think of it in a reverse manner. If we know `3` numbers such that the square of the largest number is the sum of the square of the other `2` numbers, we can qualify the `3` numbers as Pythagorean triples.
Solved Examples
Q`1`. The length of a right-angled triangle's perpendicular side (altitude) is `11` cm, and the base length is 9 cm. Determine the measurement of the hypotenuse of this right-angle triangle.
Solution: According to the provided information,
Height `= 11` cm and Base `= 9` cm
Since, `\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2`
\(\begin{align*}
\text{Hypotenuse}^2 &= 9^2 + 11^2 \\
\text{Hypotenuse}^2 &= 81 + 121 \\
\text{Hypotenuse}^2 &= 202 \\
\text{Hypotenuse} &= \sqrt{202} \\
\text{Hypotenuse} &\approx 14.21
\end{align*}\)
Thus, the length of the hypotenuse is `14.21` cm approximately.
Q`2`. Determine the length of the missing side of the provided right-angle triangle.
Solution: According to the provided information,
Height `= 5` units, and Base `= 3` units
Since, `\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2`
\(\begin{align*}
\text{Hypotenuse}^2 &= 3^2 + 5^2 \\
\text{Hypotenuse}^2 &= 9 + 25 \\
\text{Hypotenuse}^2 &= 34 \\
\text{Hypotenuse} &= \sqrt{34} \\
\text{Hypotenuse} &= 5.8
\end{align*}\)
Thus, the length of the hypotenuse is `5.8` units approximately.
Q`3`. Shelby placed a ladder from the ground to her window to set up the flower pot. The distance between wall and the bottom of the ladder is `13` m, while the window is `12` m from the ground. Determine the length of the ladder.
Solution: Let’s visualize this case through a figure (given below).
Distance between the wall and the bottom of the ladder (Base) `= 13` m
Length between ground to the window on the well (Height) `= 12` m
Length of the ladder (Hypotenuse) `=` Unknown
According to the provided information,
Height `= 12` m, and Base `= 13` m
Since, `\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2`
\(\begin{align*}
\text{Hypotenuse}^2 &= 13^2 + 12^2 \\
\text{Hypotenuse}^2 &= 169 + 144 \\
\text{Hypotenuse}^2 &= 313 \\
\text{Hypotenuse} &= \sqrt{313} \\
\text{Hypotenuse} &\approx 17.66
\end{align*}\)
Thus, the length of the ladder (hypotenuse) is `14.59` units.
Q`4`. Verify whether the given triangle is a right-angle triangle or not.
Solution: We can verify it by using the Pythagorean theorem,
`\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2`
`13^2 = 12^2 + 5^2`
`169 = 144 + 25`
`169 = 169`
Since the square of the hypotenuse is equal to the sum of the square of the other two sides, therefore it is a right angle triangle.
Q`5`. Determine the length of the base of the provided triangle.
Solution: According to the provided information,
Height `= 6` units, and Hypoteonus `= 10` units
Since, `\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2`
\(\begin{align*}
10^2 &= \text{Base}^2 + 6^2 \\
100 &= \text{Base}^2 + 36 \\
\text{Base}^2 &= 100 - 36 \\
\text{Base}^2 &= 64 \\
\text{Base} &= \sqrt{64} \\
\text{Base} &= 8
\end{align*}\)
Thus, the base of the provided triangle is `8` units.
Practice problems
Q`1`. Pythagoras theorem is applicable to the sides of _______.
- Isosceles Triangle
- Equilateral Triangle
- Right-angled Triangle
- Scalene Triangle
Answer: c
Q`2`. Which of the following is the Pythagoras formula?
- `\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2`
- `\text{Base}^2 = \text{Hypotenous}^2 + \text{Height}^2`
- `\text{Height}^2 = \text{Base}^2 + \text{Hypotenuse}^2`
- `\text{Hypotenuse}^2 = \text{Base}^2 - \text{Height}^2`
Answer: a
Q`3`. What is the missing side of the provided triangle?
- `7` units
- `4` units
- `9` units
- `5` units
Answer: b
Q`4`. Which of the following satisfies the Pythagorean theorem?
- Triangle with sides `6` units, `4` units, and `10` units
- Triangle with `4` units, `3` units, and `5` units
- Triangle with `8` units, `9` units, `10` units
- Triangle with `2` units, `8` units, `7` units
Answer: b
Q5. Pythagoras was -
- an American Mathematician
- an Indian Mathematician
- A Greek Mathematician
- A German Mathematician
Answer: c
Frequently Asked Questions
Q`1`. Does Pythagoras theorem apply to any triangle?
Answer: No, the Pythagorean theorem is applicable only for right-angle triangles.
Q`2`. What does the Pythagoras theorem state?
Answer: Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Q`3`. What is a set of three numbers called if they satisfy the Pythagoras theorem?
Answer: The set of three such numbers are called Pythagorean triples. In a Pythagorean triple, the square of the largest number is equal to the sum of the squares of the other two numbers.
Q`4`. List down some of the applications of Pythagorean theorem in real life.
Answer: There are various scenarios where one can use Pythagoras theorem. Some of them are listed below:
- We can use Pythagorean theorem to find the steepness of the hills or mountains.
- When constructing buildings with right-angled corners, builders use the theorem to ensure that the walls are perpendicular and that the corners are square. Additionally, determining the diagonal measurements of rooms are other applications where the Pythagorean theorem is employed in construction and architecture.
- In playing sports like baseball or softball, players and coaches use the theorem to calculate the distance between bases. Similarly, in recreational activities like setting up a volleyball or badminton net, knowing the diagonal distance of the court helps ensure proper placement and alignment.
- DIY enthusiasts frequently use the Pythagorean theorem in various projects around the house, such as constructing shelves, cabinets, or outdoor structures like pergolas and decks. For example, when building a pergola, the theorem helps ensure that the posts are set at right angles to each other for stability.
- When arranging furniture in a room, especially in corners or against walls, people often use the Pythagorean theorem to calculate the diagonal distance between two points. This helps ensure that furniture fits properly and that space is utilized efficiently. For instance, when placing a large sofa in a corner, knowing the diagonal distance helps determine if the sofa will fit without obstructing other furniture or pathways.