The Pythagorean Theorem applies only to right-angled triangles. It connects the lengths of the two shorter sides with the longest side, called the hypotenuse.
The square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
Pythagorean Theorem :
$$\boldsymbol{a^{2} + b^{2} = c^{2}}$$
Here, a and b are the two sides that form the 90° angle, while c is the hypotenuse.
A right-angled triangle has three sides. The two sides that meet at the 90° angle are the shorter sides, while the side opposite the 90° angle is the hypotenuse.
Note : The hypotenuse is always the side directly opposite the 90° angle. It is also always the longest side.
When the two shorter sides are known and the hypotenuse is missing, square both shorter sides, add the results, and then take the square root.
Example: A right-angled triangle has shorter sides of 3 cm and 4 cm. Find the hypotenuse.
Step 1: Write the formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 2: Substitute the known values.
$\boldsymbol{3^{2} + 4^{2} = c^{2}}$
Step 3: Square the numbers.
$\boldsymbol{9 + 16 = c^{2}}$
Step 4: Add.
$\boldsymbol{25 = c^{2}}$
Step 5: Take the square root.
$\boldsymbol{c = \sqrt{25}}$
$\boldsymbol{c = 5\text{ cm}}$
Answer: The hypotenuse is 5 cm.
The 3-4-5 triangle is one of the most common right-angled triangles. Its side lengths satisfy the Pythagorean Theorem.
Verification:
32 + 42 = 52
9 + 16 = 25
25 = 25
Therefore, 3 cm, 4 cm, and 5 cm form a right-angled triangle.
The theorem can be visualized by constructing a square on each side of the right-angled triangle.
The area of the square on the hypotenuse is equal to the combined areas of the squares on the two shorter sides.
For a 3-4-5 triangle:
Square on the 3 cm side:
$\boldsymbol{3 \times 3 = 9\text{ cm}^{2}}$
Square on the 4 cm side:
$\boldsymbol{4 \times 4 = 16\text{ cm}^{2}}$
Combined area:
$\boldsymbol{9 + 16 = 25\text{ cm}^{2}}$
Square on the 5 cm side:
$\boldsymbol{5 \times 5 = 25\text{ cm}^{2}}$
Therefore, the two smaller squares together have exactly the same area as the square built on the hypotenuse.
The Pythagorean Theorem can be rearranged to find any missing side of a right-angled triangle.
The method depends on which side is unknown. If the hypotenuse is missing, add the squared shorter sides. If one of the shorter sides is missing, subtract the known shorter side's square from the hypotenuse's square.
When the two shorter sides are known, use:
$$\boldsymbol{c = \sqrt{a^{2} + b^{2}}}$$Example: A right-angled triangle has a base of 5 cm and a height of 12 cm. Find the hypotenuse.
Step 1: Write the formula.
$\boldsymbol{c = \sqrt{a^{2} + b^{2}}}$
Step 2: Substitute the known values.
$\boldsymbol{c = \sqrt{5^{2} + 12^{2}}}$
Step 3: Square the numbers.
$\boldsymbol{c = \sqrt{25 + 144}}$
Step 4: Add.
$\boldsymbol{c = \sqrt{169}}$
Step 5: Take the square root.
$\boldsymbol{c = 13\text{ cm}}$
Answer: The hypotenuse is 13 cm.
When the hypotenuse and one shorter side are known, rearrange the theorem by subtracting the known shorter side's square from the hypotenuse's square.
$$\boldsymbol{a = \sqrt{c^{2} - b^{2}}}$$ $$\boldsymbol{b = \sqrt{c^{2} - a^{2}}}$$A right-angled triangle has a base of 6 m and a hypotenuse of 10 m. Find the missing height.
Step 1: Write the formula.
$\boldsymbol{b = \sqrt{c^{2} - a^{2}}}$
Step 2: Substitute the known values.
$\boldsymbol{b = \sqrt{10^{2} - 6^{2}}}$
Step 3: Square the numbers.
$\boldsymbol{b = \sqrt{100 - 36}}$
Step 4: Subtract.
$\boldsymbol{b = \sqrt{64}}$
Step 5: Take the square root.
$\boldsymbol{b = 8\text{ m}}$
Answer: The missing height is 8 m.
The same method works when the other shorter side is missing. Subtract the known shorter side squared from the hypotenuse squared, then take the square root.
Example: A right-angled triangle has a height of 9 cm and a hypotenuse of 15 cm. Find the missing base.
Step 1: Write the formula.
$\boldsymbol{a = \sqrt{c^{2} - b^{2}}}$
Step 2: Substitute the known values.
$\boldsymbol{a = \sqrt{15^{2} - 9^{2}}}$
Step 3: Square the numbers.
$\boldsymbol{a = \sqrt{225 - 81}}$
Step 4: Subtract.
$\boldsymbol{a = \sqrt{144}}$
Step 5: Take the square root.
$\boldsymbol{a = 12\text{ cm}}$
Answer: The missing base is 12 cm.
The easiest way to choose the formula is to first identify the missing side.
Missing hypotenuse:
$$\boldsymbol{c = \sqrt{a^{2} + b^{2}}}$$Add the squares of the two shorter sides.
Missing shorter side:
$$\boldsymbol{a = \sqrt{c^{2} - b^{2}}}$$ $$\boldsymbol{b = \sqrt{c^{2} - a^{2}}}$$Subtract the known shorter side squared from the hypotenuse squared.
After finding a missing side, you can substitute all three side lengths back into the Pythagorean Theorem to check whether they work.
For the 6-8-10 triangle:
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
$\boldsymbol{6^{2} + 8^{2} = 10^{2}}$
$\boldsymbol{36 + 64 = 100}$
$\boldsymbol{100 = 100}$
Therefore, the answer is correct.
Never add the squared values when finding a missing shorter side. The hypotenuse is always the largest side, so its squared value must be the starting value when solving for either shorter side.
Hypotenuse missing → ADD
Shorter side missing → SUBTRACT
The Pythagorean Theorem is useful when a problem involves a diagonal, slanted distance, height, or missing length that creates a right-angled triangle.
A good first step is to look for a 90° angle. If the required distance forms a right triangle, the Pythagorean Theorem can usually be applied.
A rectangle does not contain a visible right-angled triangle until a diagonal is drawn from one corner to the opposite corner.
The diagonal divides the rectangle into two congruent right-angled triangles. The length and width become the two shorter sides, while the diagonal becomes the hypotenuse.
Example: A rectangular board is 15 cm long and 8 cm wide. Find the length of its diagonal.
Step 1: Identify the right triangle.
The diagonal divides the rectangle into two right-angled triangles. Therefore:
a = 15 cm
b = 8 cm
c = d = ?
Step 2: Apply the Pythagorean Theorem.
$\boldsymbol{d^{2} = 15^{2} + 8^{2}}$
Step 3: Square the known lengths and add them.
$\boldsymbol{d^{2} = 225 + 64}$
$\boldsymbol{d^{2} = 289}$
Step 4: Find the square root.
$\boldsymbol{d = \sqrt{289}}$
$\boldsymbol{d = 17\text{ cm}}$
Answer: The diagonal is 17 cm.
A square has four equal sides. When a diagonal is drawn from one corner to the opposite corner, it creates two right-angled triangles.
Since both shorter sides have the same length, the theorem becomes:
A square has a side length of 10 cm. Find its diagonal.
Solution :
The diagonal divides the square into two right-angled triangles.
Let s represent the side of the square and d the diagonal of the square.
$\boldsymbol{s = 10\text{ cm}}$
$\boldsymbol{d = ?}$
Step 2: Apply the Pythagorean Theorem.
$\boldsymbol{d^{2} = s^{2} + s^{2}}$
Step 3: Square the sides of the square and add them.
$\boldsymbol{d^{2} = 10^{2} + 10^{2}}$
$\boldsymbol{d^{2} = 100 + 100}$
$\boldsymbol{d^{2} = 200}$
Step 4: Find the square root.
$\boldsymbol{d = \sqrt{200}}$
$\boldsymbol{d = 14.14\text{ cm}}$
Answer: The diagonal is 14.14 cm.
Note: This answer has been rounded to 2 decimal places. Therefore , the resulting decimal value is not perfectly accurate.
A ladder leaning against a wall forms a right-angled triangle with the wall and the ground.
The ladder is the hypotenuse because it is opposite the 90° angle.
Example: A 13 m ladder is placed against a wall. Its base is 5 m from the wall. How high up the wall does the ladder reach?
Step 1: Identify the known sides.
Hypotenuse = 13 m
Base = 5 m
Height = h
Step 2: Use the theorem.
$\boldsymbol{h^{2} + 5^{2} = 13^{2}}$
Step 3: Square the known values.
$\boldsymbol{h^{2} + 25 = 169}$
Step 4: Subtract 25.
$\boldsymbol{h^{2} = 169 - 25}$
$\boldsymbol{h^{2} = 144}$
Step 5: Find the square root.
$\boldsymbol{h = \sqrt{144}}$
$\boldsymbol{h = 12\text{ m}}$
Answer: The ladder reaches 12 m up the wall.
A Pythagorean triple consists of three positive integers (a, b, c) that perfectly satisfy the Pythagorean theorem formula $\boldsymbol{a^{2} + b^{2} = c^{2}}$.
Pythagorean triples are extremely helpful because they allow us to instantly know if a set of three given dimensions can form a perfect right-angled triangle without drawing it. If the square of the largest dimension equals the sum of the squares of the two smaller dimensions, the dimensions form a right-angled triangle.
Here are the most frequently used basic Pythagorean triples in mathematics:
If you multiply or divide all three dimensions in a Pythagorean triple by the same positive whole number, the resulting dimensions will also form a valid right-angled triangle.
(i) Multiplying (3, 4, 5) by 2 gives (6, 8, 10). These dimensions form a right-angled triangle because $\boldsymbol{6^{2} + 8^{2} = 36 + 64 = 100 = 10^{2}}$.
(ii) Multiplying (3, 4, 5) by 3 gives (9, 12, 15). These dimensions form a right-angled triangle because $\boldsymbol{9^{2} + 12^{2} = 81 + 144 = 225 = 15^{2}}$.
(iii) Multiplying (5, 12, 13) by 2 gives (10, 24, 26). These dimensions form a right-angled triangle because $\boldsymbol{10^{2} + 24^{2} = 100 + 576 = 676 = 26^{2}}$.
Verify whether a triangle with dimensions 8 cm, 15 cm, and 17 cm forms a right-angled triangle.
Step 1: Identify the smaller dimensions as a and b, and the largest dimension as c.
a = 8 cm
b = 15 cm
c = 17 cm
Step 2: State the Pythagorean theorem verification formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 3: Substitute the dimensions into the formula.
$\boldsymbol{8^{2} + 15^{2} = 17^{2}}$
Step 4: Square all three individual numbers.
$\boldsymbol{64 + 225 = 289}$
Step 5: Add the two numbers on the left side together.
$\boldsymbol{289 = 289}$
Step 6: Compare both sides of the equation.
Since both sides are perfectly equal, these dimensions satisfy the formula.
Final Answer: Yes, the dimensions form a right-angled triangle.
An Isosceles triangle has two equal sides, while an Equilateral triangle has three equal sides. Neither of these triangles contains a right angle naturally.
However, by drawing a perpendicular height line from the top vertex straight down to the base, we cut these triangles exactly in half. This action creates two identical right-angled triangles, allowing us to easily find any missing dimension using the Pythagorean theorem.
An isosceles triangle has two sides of equal length. By drawing a perpendicular line straight down from the top vertex to the base, we divide it into two identical right-angled triangles. This lets us use the Pythagorean theorem to calculate the missing height or side.
An isosceles triangle has two equal sides of length 10 cm and a base of 12 cm. Calculate its perpendicular height (h).
Step 1: Divide the full base length by 2 to find the base of the right-angled triangle.
$\boldsymbol{a = 12\text{ cm} \div 2}$
$\boldsymbol{a = 6\text{ cm}}$
Step 2: Identify the known hypotenuse and the unknown side.
Hypotenuse $\boldsymbol{c = }$ 10 cm
$\boldsymbol{b}$ (Height) = ?
Step 3: State the Pythagorean theorem formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 4: Substitute the values into the formula.
$\boldsymbol{6^{2} + b^{2} = 10^{2}}$
Step 5: Square both individual numbers.
$\boldsymbol{36 + b^{2} = 100}$
Step 6: Isolate $\boldsymbol{b^{2}}$ by subtracting 36 from 100.
$\boldsymbol{b^{2} = 100 - 36}$
Step 7: Subtract the numbers.
$\boldsymbol{b^{2} = 64}$
Step 8: Take the square root of both sides.
$\boldsymbol{b = \sqrt{64}}$
Step 9: Calculate the value of $\boldsymbol{b}$.
$\boldsymbol{b = 8}$
Final Answer: Height = 8 cm
An equilateral triangle has three sides of equal length. Splitting it straight down the middle from the top vertex creates a perpendicular line that bisects the base, enabling us to calculate its height using the Pythagorean theorem.
An equilateral triangle has a side length of 6 m. Calculate its perpendicular height (h), leaving your answer under a square root sign.
Step 1: Divide the bottom base side by 2 to find the base of the half triangle.
$\boldsymbol{a = 6\text{ m} \div 2}$
$\boldsymbol{a = 3\text{ m}}$
Step 2: Identify the known hypotenuse value and the unknown side.
Hypotenuse $\boldsymbol{c = }$ 6 m
$\boldsymbol{b}$ (Height) = ?
Step 3: State the Pythagorean theorem formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 4: Substitute the values into the formula.
$\boldsymbol{3^{2} + b^{2} = 6^{2}}$
Step 5: Square both individual numbers.
$\boldsymbol{9 + b^{2} = 36}$
Step 6: Isolate $\boldsymbol{b^{2}}$ by subtracting 9 from 36.
$\boldsymbol{b^{2} = 36 - 9}$
Step 7: Subtract the numbers.
$\boldsymbol{b^{2} = 27}$
Step 8: Take the square root of both sides.
$\boldsymbol{b = \sqrt{27}}$
$\boldsymbol{b = 5.20}$
Final Answer: Height = 5.20 m
Note: This answer has been rounded to 2 decimal places. Therefore , the resulting decimal value is not perfectly accurate.
A right-angled triangle has sides of lengths 3 cm and 4 cm. Find the length of the hypotenuse.
Step 1: Identify the given sides and the unknown side.
$\boldsymbol{a = 3\text{ cm}}$
$\boldsymbol{b = 4\text{ cm}}$
$\boldsymbol{c}$ (Hypotenuse) = ?
Step 2: State the Pythagorean theorem formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 3: Substitute the known values into the formula.
$\boldsymbol{3^{2} + 4^{2} = c^{2}}$
Step 4: Square the values of the sides.
$\boldsymbol{9 + 16 = c^{2}}$
Step 5: Add the squared values together.
$\boldsymbol{25 = c^{2}}$
Step 6: Take the square root of both sides to find c.
$\boldsymbol{c = \sqrt{25}}$
Step 7: Calculate the value of c.
$\boldsymbol{c = 5}$
Final Answer: Hypotenuse = 5 cm
The hypotenuse of a right-angled triangle is 13 m long and one of its sides is 5 m. Find the length of the other side.
Step 1: Identify the given sides and the unknown side.
$\boldsymbol{a = 5\text{ m}}$
$\boldsymbol{c}$ (Hypotenuse) = 13 m
$\boldsymbol{b} = ?$
Step 2: State the Pythagorean theorem formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 3: Substitute the known values into the formula.
$\boldsymbol{5^{2} + b^{2} = 13^{2}}$
Step 4: Square the known side and the hypotenuse.
$\boldsymbol{25 + b^{2} = 169}$
Step 5: Isolate b² by subtracting 25 from 169.
$\boldsymbol{b^{2} = 169 - 25}$
Step 6: Subtract the numbers to get the value of b².
$\boldsymbol{b^{2} = 144}$
Step 7: Take the square root to find b.
$\boldsymbol{b = \sqrt{144}}$
Step 8: Calculate the value of b.
$\boldsymbol{b = 12}$
Final Answer = 12 m
A ladder is leaning against a vertical wall. The foot of the ladder is 6 feet away from the base of the wall, and the ladder reaches 8 feet up the wall. Find the length of the ladder.
Step 1: Identify the given sides and the unknown side.
$\boldsymbol{a}$ (Distance from wall) = 6 feet
$\boldsymbol{b}$ (Height up the wall) = 8 feet
$\boldsymbol{c}$ (Length of the ladder) = ?
Step 2: State the Pythagorean theorem formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 3: Substitute the known values into the formula.
$\boldsymbol{6^{2} + 8^{2} = c^{2}}$
Step 4: Square both sides.
$\boldsymbol{36 + 64 = c^{2}}$
Step 5: Add the squared numbers together.
$\boldsymbol{100 = c^{2}}$
Step 6: Take the square root to find c.
$\boldsymbol{c = \sqrt{100}}$
Step 7: Calculate the value of c.
$\boldsymbol{c = 10}$
Final Answer: Length of the ladder = 10 feet
A rectangular field has a length of 15 meters and a width of 8 meters. Calculate the length of the diagonal path connecting two opposite corners.
Step 1: Identify the right-angled triangle formed by the length, width, and diagonal.
$\boldsymbol{a}$ (Width) = 8 m
$\boldsymbol{b}$ (Length) = 15 m
$\boldsymbol{c}$ (Diagonal) = ?
Step 2: State the Pythagorean theorem formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 3: Substitute the width and length into the formula.
$\boldsymbol{8^{2} + 15^{2} = c^{2}}$
Step 4: Square the width and the length.
$\boldsymbol{64 + 225 = c^{2}}$
Step 5: Add the two squared values together.
$\boldsymbol{289 = c^{2}}$
Step 6: Take the square root of the sum to find c.
$\boldsymbol{c = \sqrt{289}}$
Step 7: Calculate the value of c.
$\boldsymbol{c = 17}$
Final Answer: Length of the diagonal = 17 meters
An isosceles triangle has two equal sides of length 10 cm and a base of length 12 cm. Find the perpendicular height of the triangle.
Step 1: Divide the isosceles triangle into two identical right-angled triangles where the perpendicular height bisects the base.
$\boldsymbol{a}$ (Base of right triangle) = $\boldsymbol{12 \div 2 = }$ 6 cm
$\boldsymbol{c}$ (Hypotenuse) = 10 cm
$\boldsymbol{b}$ (Perpendicular height) = ?
Step 2: State the Pythagorean theorem formula.
$\boldsymbol{a^{2} + b^{2} = c^{2}}$
Step 3: Substitute the half-base and hypotenuse values into the formula.
$\boldsymbol{6^{2} + b^{2} = 10^{2}}$
Step 4: Square the side and the hypotenuse numbers.
$\boldsymbol{36 + b^{2} = 100}$
Step 5: Isolate b² by subtracting 36 from 100.
$\boldsymbol{b^{2} = 100 - 36}$
Step 6: Subtract the values to get b².
$\boldsymbol{b^{2} = 64}$
Step 7: Take the square root to find b.
$\boldsymbol{b = \sqrt{64}}$
Step 8: Calculate the value of b.
$\boldsymbol{b = 8}$
Final Answer: Perpendicular height = 8 cm
Attempt the following Questions:
(i) A right-angled triangle has a base of 9 cm and a height of 12 cm. Calculate the length of the hypotenuse.
(ii) The hypotenuse of a right-angled triangle is 26 m long and its height is 10 m. Determine the length of the base.
(iii) A ladder is leaning against a vertical wall. The foot of the ladder is 7 feet away from the base of the wall, and the ladder itself is 25 feet long. Calculate how high up the wall the ladder reaches.
(iv) A rectangular field has a length of 15 meters and a width of 8 meters. Find the length of the diagonal path connecting two opposite corners.
(v) A ship sails 12 miles due north and then 5 miles due east. Determine the direct straight-line distance from the ship's starting point to its final position.
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