129. Right-Angled Triangles, Hypotenuse and Pythagorean Triples

Learning Intentions

• Identify the hypotenuse in a right-angled triangle
• Determine if three numbers form a Pythagorean triple
• Use Pythagoras’ theorem to determine if a triangle has a right angle based on its side lengths

Pre-requisite Summary

• Know that a right angle is
• Be able to identify a right-angled triangle from a diagram
• Know that the side opposite the right angle is special in a right-angled triangle
• Be able to square whole numbers
• Understand that a theorem is a mathematical rule that can be used to test a statement
• Be able to compare two numerical expressions to see if they are equal

Worked Examples

Worked Example 1

Identify the hypotenuse in each right-angled triangle:

a) a triangle with sides labelled , , , where is opposite the right angle

b) a triangle with side lengths , , , where the right angle is between the sides of lengths and

Worked Example 2

Determine whether each set of numbers is a Pythagorean triple:

a)

b)

c)

Worked Example 3

Determine whether each set of numbers is a Pythagorean triple:

a)

b)

c)

Worked Example 4

Use Pythagoras’ theorem to determine whether a triangle with side lengths , and has a right angle.

Worked Example 5

Use Pythagoras’ theorem to determine whether a triangle with side lengths , and has a right angle.

Worked Example 6

For each triangle, decide whether it is right-angled:

a) side lengths , ,

b) side lengths , ,

c) side lengths , ,

Problems

Problem 1

Identify the hypotenuse in each right-angled triangle:

a) a triangle with sides labelled , , , where is opposite the right angle

b) a triangle with side lengths , , , where the right angle is between the sides of lengths and

Problem 2

Determine whether each set of numbers is a Pythagorean triple:

a)

b)

c)

Problem 3

Determine whether each set of numbers is a Pythagorean triple:

a)

b)

c)

Problem 4

Use Pythagoras’ theorem to determine whether a triangle with side lengths , and has a right angle.

Problem 5

Use Pythagoras’ theorem to determine whether a triangle with side lengths , and has a right angle.

Problem 6

For each triangle, decide whether it is right-angled:

a) side lengths , ,

b) side lengths , ,

c) side lengths , ,

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement:

a) In a right-angled triangle, the hypotenuse is the side opposite the ______ angle

b) The hypotenuse is always the ______ side in a right-angled triangle

c) Pythagoras’ theorem is ______

Exercise 2.

Identify the hypotenuse in each triangle:

a) sides , , , with the right angle between and

b) sides , , , with the right angle between and

c) sides labelled , , , where is opposite the right angle

Exercise 3.

Determine whether each set of numbers is a Pythagorean triple:

a)

b)

c)

d)

Exercise 4.

Determine whether each set of numbers is a Pythagorean triple:

a)

b)

c)

d)

Exercise 5.

Use Pythagoras’ theorem to decide whether each triangle is right-angled:

a) side lengths , ,

b) side lengths , ,

c) side lengths , ,

Exercise 6.

Use Pythagoras’ theorem to decide whether each triangle is right-angled:

a) side lengths , ,

b) side lengths , ,

c) side lengths , ,

Exercise 7.

For each set of side lengths, identify the longest side first, then test for a right angle:

a)

b)

c)

Exercise 8.

Decide whether each statement is true or false:

a) Every set of three even numbers is a Pythagorean triple

b) In a right-angled triangle, the hypotenuse can be one of the shorter sides

c) If , then the triangle is right-angled

d) The hypotenuse is opposite the right angle

Reasoning

Exercise 9.

Explain why the hypotenuse must always be the longest side in a right-angled triangle.

Exercise 10.

A student says that is not a Pythagorean triple because it is not . Explain the mistake.

Exercise 11.

Noah says that to test whether a triangle is right-angled, you can square any two sides and add them. Is he correct? Explain.

Exercise 12.

Explain why the longest side must be used as the value of when checking .

Exercise 13.

A student tests the numbers by writing . Describe the error.

Problem-solving

Exercise 14.

A triangular sign has side lengths cm, cm and cm. Determine whether it has a right angle.

Exercise 15.

A ladder, the ground and a wall form a triangle with side lengths m, m and m. Is the corner between the wall and ground a right angle?

Exercise 16.

A builder measures a triangular frame with side lengths cm, cm and cm. Determine whether the frame is right-angled.

Exercise 17.

A right-angled triangle has side lengths cm, cm and cm. Identify the hypotenuse.

Exercise 18.

A student claims that a triangle with side lengths cm, cm and cm is right-angled. Check whether the claim is correct.

Exercise 19.

A surveyor measures three sides of a triangular plot as m, m and m. Determine whether the plot contains a right angle.

Potential Misunderstandings

• Students may think the hypotenuse is any side that touches the right angle
• Students may forget that the hypotenuse is opposite the right angle, not beside it
• Students may not identify the longest side before testing Pythagoras’ theorem
• Students may use the wrong side as in
• Students may think only the triple works and not Recognise multiples such as
• Students may assume a triangle is right-angled just because the side lengths “look close”
• Students may make squaring errors when checking a Pythagorean triple
• Students may think that any three numbers with an even number among them form a Pythagorean triple
• Students may reverse Pythagoras’ theorem incorrectly when testing side lengths

Next: 130. Pythagoras and Finding the Hypotenuse