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 ${90}^{\circ }$
• 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 $a$, $b$, $c$, where $c$ is opposite the right angle
b) a triangle with side lengths $5$, $12$, $13$, where the right angle is between the sides of lengths $5$ and $12$

Worked Example 2

Determine whether each set of numbers is a Pythagorean triple:
a) $3,4,5$
b) $5,12,13$
c) $6,8,11$

Worked Example 3

Determine whether each set of numbers is a Pythagorean triple:
a) $8,15,17$
b) $7,24,25$
c) $9,10,12$

Worked Example 4

Use Pythagoras’ theorem to determine whether a triangle with side lengths $6$, $8$ and $10$ has a right angle.

Worked Example 5

Use Pythagoras’ theorem to determine whether a triangle with side lengths $7$, $9$ and $12$ has a right angle.

Worked Example 6

For each triangle, decide whether it is right-angled:
a) side lengths $9$, $12$, $15$
b) side lengths $10$, $24$, $26$
c) side lengths $4$, $5$, $6$

Problems

Problem 1

Identify the hypotenuse in each right-angled triangle:
a) a triangle with sides labelled $x$, $y$, $z$, where $z$ is opposite the right angle
b) a triangle with side lengths $8$, $15$, $17$, where the right angle is between the sides of lengths $8$ and $15$

Problem 2

Determine whether each set of numbers is a Pythagorean triple:
a) $9,12,15$
b) $10,24,26$
c) $5,6,8$

Problem 3

Determine whether each set of numbers is a Pythagorean triple:
a) $12,16,20$
b) $20,21,29$
c) $8,10,13$

Problem 4

Use Pythagoras’ theorem to determine whether a triangle with side lengths $5$, $12$ and $13$ has a right angle.

Problem 5

Use Pythagoras’ theorem to determine whether a triangle with side lengths $6$, $7$ and $9$ has a right angle.

Problem 6

For each triangle, decide whether it is right-angled:
a) side lengths $12$, $35$, $37$
b) side lengths $11$, $13$, $16$
c) side lengths $16$, $30$, $34$

Exercises

Understanding and Fluency

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 $a^2+b^2=$ ______

2. Identify the hypotenuse in each triangle:
   a) sides $3$, $4$, $5$, with the right angle between $3$ and $4$
   b) sides $8$, $6$, $10$, with the right angle between $6$ and $8$
   c) sides labelled $p$, $q$, $r$, where $r$ is opposite the right angle

3. Determine whether each set of numbers is a Pythagorean triple:
   a) $3,4,5$
   b) $5,12,13$
   c) $7,8,10$
   d) $8,15,17$

4. Determine whether each set of numbers is a Pythagorean triple:
   a) $9,40,41$
   b) $6,8,10$
   c) $10,11,15$
   d) $20,21,29$

5. Use Pythagoras’ theorem to decide whether each triangle is right-angled:
   a) side lengths $8$, $15$, $17$
   b) side lengths $9$, $10$, $14$
   c) side lengths $7$, $24$, $25$

6. Use Pythagoras’ theorem to decide whether each triangle is right-angled:
   a) side lengths $12$, $16$, $20$
   b) side lengths $14$, $18$, $22$
   c) side lengths $11$, $60$, $61$

7. For each set of side lengths, identify the longest side first, then test for a right angle:
   a) $4,13,12$
   b) $15,36,39$
   c) $18,24,30$

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 $a^2+b^2=c^2$, then the triangle is right-angled
   d) The hypotenuse is opposite the right angle

Reasoning

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

10. A student says that $6,8,10$ is not a Pythagorean triple because it is not $3,4,5$. Explain the mistake.

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.

12. Explain why the longest side must be used as the value of $c$ when checking $a^2+b^2=c^2$.

13. A student tests the numbers $5,12,13$ by writing ${13}^2+{12}^2=5^2$. Describe the error.

Problem-solving

14. A triangular sign has side lengths $9$ cm, $12$ cm and $15$ cm. Determine whether it has a right angle.

15. A ladder, the ground and a wall form a triangle with side lengths $6$ m, $8$ m and $10$ m. Is the corner between the wall and ground a right angle?

16. A builder measures a triangular frame with side lengths $20$ cm, $21$ cm and $29$ cm. Determine whether the frame is right-angled.

17. A right-angled triangle has side lengths $7$ cm, $24$ cm and $25$ cm. Identify the hypotenuse.

18. A student claims that a triangle with side lengths $10$ cm, $10$ cm and $14$ cm is right-angled. Check whether the claim is correct.

19. A surveyor measures three sides of a triangular plot as $12$ m, $35$ m and $37$ 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 $c$ in $a^2+b^2=c^2$
• Students may think only the triple $3,4,5$ works and not recognise multiples such as $6,8,10$
• 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