# Identify Right-Angled Triangles

In this worksheet, students will use the converse of Pythagoras' Theorem to determine whether or not a triangle is right-angled.

Key stage:  KS 3

Curriculum topic:   Geometry and Measures

Curriculum subtopic:   Apply Facts About Angles and Sides

Difficulty level:

### QUESTION 1 of 10

Pythagorean theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Thus in the above right-angled triangle a2 + b2  = c2

If a triangle of sides a, b and c is such that a2 + b2 = c2, then the triangle is right-angled.

Example 1

A triangle has sides of length 6 cm, 8 cm and 9 cm.

Is it right-angled?

First, pick the two shorter sides, square them and then add:

62 + 82 = 36 + 64 = 100

Then square the longest side:

92 = 81

So 62 + 82 ≠ 92 and the triangle is therefore not right-angled.

Example 2

A triangle has sides of length 5 cm, 13 cm and 12 cm.

Is it right-angled?

First, pick the two shorter sides, square them and then add:

52 + 122 = 25 + 144 = 169

Then square the longest side:

132 = 169

So 52 + 122 = 132 and the triangle is therefore right-angled.

Don't worry too much if you feel unsure or overloaded with information.

We will work through ten questions together so that you feel super confident!

A triangle has sides of length 3 cm, 4 cm and 3 cm.

Is it right-angled?

Yes

No

A triangle has sides of the length of 5 cm, 3 cm and 4 cm.

Is it right-angled?

Yes

No

A triangle has sides of the length of 10 cm, 13 cm and 12 cm.

How would we work out if the triangle is right-angled?

Yes

No

A triangle has sides of length 10 cm, 26 cm and 17 cm.

In this instance, what is a² + b²?

189 cm

100 cm

289 cm

389 cm

A triangle has sides of the length of 10 cm, 26 cm and 24 cm.

Is it right-angled?

Yes

No

A triangle has sides of length 10 cm, 7 cm and 8 cm.

What is c²?

49 cm

84 cm

64 cm

100 cm

A triangle has sides of length 10 cm, 7 cm and 16 cm.

Is it right-angled?

Yes

No

A triangle has sides of length 15 cm, 17 cm and 8 cm.

Is it right-angled?

Yes

No

A triangle has sides of length 45 cm, 37 cm and 11 cm.

Is it right-angled?

Yes

No

A triangle has sides of the length of 40 cm, 9 cm and 41 cm.

What is a² + b²?

1600 cm

1681 cm

1581 cm

1400 cm

• Question 1

A triangle has sides of length 3 cm, 4 cm and 3 cm.

Is it right-angled?

No
EDDIE SAYS
First, pick the two shorter sides, square them and then add: (3 x 3) + (3 x 3) = 18 Then square the longest side: 4 x 4 = 16 (3 x 3) + (3 x 3) does not equal (4 x 4). Therefore, the triangle is not right-angled.
• Question 2

A triangle has sides of the length of 5 cm, 3 cm and 4 cm.

Is it right-angled?

Yes
EDDIE SAYS
As you did with the previous question we square the shortest sides and add them together: = (3 x 3) + (4 x 4) = 9 + 16 = 25 Then, we square the longest side (5 x 5) = 25. This triangle is squared as a² + b² = c² Great focus, let's push on.
• Question 3

A triangle has sides of the length of 10 cm, 13 cm and 12 cm.

How would we work out if the triangle is right-angled?

EDDIE SAYS
If you wanted to push yourself you could have done the working out yourself. a² = 10² = 100 b² = 12² = 144 100 + 144 = 244 c² = 169 As 244 and 169 are not equal, the triangle in question does not have a right angle.
• Question 4

A triangle has sides of length 10 cm, 26 cm and 17 cm.

In this instance, what is a² + b²?

389 cm
EDDIE SAYS
a² = 10² = 100 cm b² = 17² = 289 cm a² + b² = 389 cm
• Question 5

A triangle has sides of the length of 10 cm, 26 cm and 24 cm.

Is it right-angled?

Yes
EDDIE SAYS
First, pick the two shorter sides, square them and then add: (10 x 10) + (24 x 24) = 100 + 576 = 676 cm Then square the longest side: 26 x 26 = 676 cm As a² + b² = c² the triangle is right-angled. You're making tremendous strides in progress!
• Question 6

A triangle has sides of length 10 cm, 7 cm and 8 cm.

What is c²?

100 cm
EDDIE SAYS
You know that 'c' refers to the longest side. As the longest side of our triangle is 10 cm, we need to calculate 10² (or 10 x 10) which is 100 cm. Did you find this question easier? Hopefully, as you only did a fraction of the working out you will need to work out whether the triangle in question had a right-angle or not.
• Question 7

A triangle has sides of length 10 cm, 7 cm and 16 cm.

Is it right-angled?

No
EDDIE SAYS
First, we square the two shortest sides: 7² = 49 10² = 100 49 + 100 = 149 Then we square the longest side (c): 16² = 256 As a² + b² does not equal c², the triangle in question does not have a right angle.
• Question 8

A triangle has sides of length 15 cm, 17 cm and 8 cm.

Is it right-angled?

Yes
EDDIE SAYS
First, we square the two shortest sides: 8² = 64 15² = 225 64 + 225 = 289 Then we square the longest side (c): 16² = 289 As a² + b² does equal c², the triangle in question is right-angled!
• Question 9

A triangle has sides of length 45 cm, 37 cm and 11 cm.

Is it right-angled?

No
EDDIE SAYS
First, we square the two shortest sides: 11² = 121 37² = 1,369 121 + 1369 = 1,490 Then we square the longest side (c): 45² = 2,025 As a² + b² does not equal c², the triangle in question does not have a right angle. Great work if you spotted that!
• Question 10

A triangle has sides of the length of 40 cm, 9 cm and 41 cm.

What is a² + b²?

1681 cm
EDDIE SAYS
a² = 9² = 81 cm b² = 40² = 1600 cm 81 + 1600 = 1681 cm Well done that's another activity ticked off!
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