Strand 2 — Geometry & Trigonometry

The Theorem of Pythagoras and its Converse

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to state the Theorem of Pythagoras.
✓By the end of this lesson students will be able to apply the Theorem of Pythagoras to find the length of an unknown side in a right-angled triangle.
✓By the end of this lesson students will be able to solve practical problems involving the Theorem of Pythagoras.
✓By the end of this lesson students will be able to state the Converse of the Theorem of Pythagoras.
✓By the end of this lesson students will be able to use the Converse of the Theorem of Pythagoras to determine if a triangle is right-angled.

Key concepts

Theorem of Pythagoras

The Theorem of Pythagoras is a fundamental principle in geometry that applies specifically to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

a² + b² = c²

Hypotenuse

In a right-angled triangle, the hypotenuse is the longest side. It is always the side directly opposite the right angle (90°). In the formula a² + b² = c², 'c' represents the hypotenuse.

Converse of the Theorem of Pythagoras

The Converse of the Theorem of Pythagoras allows us to determine if a triangle is a right-angled triangle. It states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right-angled triangle.

If a² + b² = c², then the triangle is right-angled.

Key facts to remember

1The Theorem of Pythagoras applies only to right-angled triangles.
2The hypotenuse is the longest side of a right-angled triangle and is opposite the right angle.
3The formula is a² + b² = c², where c is the length of the hypotenuse and a and b are the lengths of the other two sides.
4To find the hypotenuse, you add the squares of the other two sides.
5To find a shorter side, you subtract the square of the known shorter side from the square of the hypotenuse.
6The Converse of the Theorem of Pythagoras states that if a² + b² = c² for a triangle with sides a, b, and c (where c is the longest side), then the triangle is a right-angled triangle.
7Always remember to take the square root as the final step to find the side length.
8Units must be consistent and included in the final answer.

Worked examples

Example 1

A right-angled triangle has sides of length 5 cm and 12 cm. Find the length of its hypotenuse.

IIdentify the known shorter sides as a = 5 cm and b = 12 cm. Let the hypotenuse be c.

IIApply the Theorem of Pythagoras: a² + b² = c²

IIISubstitute the values: 5² + 12² = c²

IVCalculate the squares: 25 + 144 = c²

VAdd the values: 169 = c²

VITake the square root of both sides: c = √169

VIICalculate the square root: c = 13

Answer

The length of the hypotenuse is 13 cm.

Example 2

The hypotenuse of a right-angled triangle is 10 m long. One of the other sides is 6 m long. Find the length of the third side.

IIdentify the hypotenuse as c = 10 m and one shorter side as a = 6 m. Let the unknown shorter side be b.

IIApply the Theorem of Pythagoras: a² + b² = c²

IIISubstitute the values: 6² + b² = 10²

IVCalculate the squares: 36 + b² = 100

VIsolate b²: b² = 100 - 36

VISubtract the values: b² = 64

VIITake the square root of both sides: b = √64

VIIICalculate the square root: b = 8

Answer

The length of the third side is 8 m.

Example 3

A ladder 6.5 m long is leaning against a vertical wall. The base of the ladder is 2.5 m from the base of the wall. How high up the wall does the ladder reach?

IDraw a diagram representing the wall, the ground, and the ladder. This forms a right-angled triangle.

IIIdentify the hypotenuse (ladder) as c = 6.5 m. Identify one shorter side (distance from wall) as a = 2.5 m. Let the height up the wall be b.

IIIApply the Theorem of Pythagoras: a² + b² = c²

IVSubstitute the values: 2.5² + b² = 6.5²

VCalculate the squares: 6.25 + b² = 42.25

VIIsolate b²: b² = 42.25 - 6.25

VIISubtract the values: b² = 36

VIIITake the square root of both sides: b = √36

9Calculate the square root: b = 6

Answer

The ladder reaches 6 m high up the wall.

Always draw a diagram for word problems to visualise the right-angled triangle and correctly identify the hypotenuse.

Example 4

A triangle has sides of length 7 cm, 24 cm, and 25 cm. Is it a right-angled triangle?

IIdentify the longest side as c = 25 cm. Let the other sides be a = 7 cm and b = 24 cm.

IICalculate the sum of the squares of the two shorter sides: a² + b² = 7² + 24²

IIICalculate the squares: 49 + 576

IVAdd the values: 625

VCalculate the square of the longest side: c² = 25²

VICalculate the square: 625

VIICompare the results: Since a² + b² = 625 and c² = 625, then a² + b² = c².

VIIIApply the Converse of the Theorem of Pythagoras.

Answer

Yes, the triangle is a right-angled triangle because 7² + 24² = 25².

Make sure to clearly state your conclusion based on the Converse of the Theorem of Pythagoras.

Common mistakes

✗Incorrectly identifying the hypotenuse (it is always opposite the right angle and is the longest side).
✗Adding the squares of the two shorter sides when trying to find a shorter side (it should be subtraction).
✗Forgetting to take the square root at the end of the calculation, leaving the answer as a squared value.
✗Applying the Theorem of Pythagoras to triangles that are not right-angled.
✗Mixing up the Theorem and its Converse, or not knowing when to use each.

Exam tips

★Always draw a clear diagram, especially for word problems, and label the sides (a, b, c) correctly.
★Clearly state the formula a² + b² = c² at the beginning of your solution.
★Show all steps of your calculation, including squaring numbers and taking the square root.
★Check your answer: the hypotenuse must always be the longest side. If your calculation gives a shorter side that is longer than the hypotenuse, you've made a mistake.
★When using the Converse, clearly show the calculation for a² + b² and c² separately before comparing them and stating your conclusion.

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