Expected Learning Outcomes

You are expected to:

Recognize the sides of a right-angled triangle in different situations
Identify the Pythagorean relationship in different situations
Apply the Pythagorean relationship to real-life situations
Promote the use of Pythagoras’ Theorem in real-life situations

Pythagorean Relationship – Grade 7 Mathematics

3.0 Measurements

3.1 Pythagorean Relationship

About 2000 years ago, an amazing discovery was made about right-angled triangles. If squares of length equivalent to the sides of the triangle are drawn, then the sum of the area of the two small squares is exactly the same as the area of the large square.

The Pythagoras theorem states that a² + b² = c², where a and b are the shorter sides while c is the longest side. The longest side is known as the hypotenuse.

Example 3.1

Solve for the value of x in the triangle shown below.

From Pythagoras theorem 5² + 12² = x²
25 + 144 = x²
169 = x²
We now take square root both sides x = 13

Example 3.2

Find out which of the two triangles drawn below is right angled.

We start by testing (a)
Is (8² + 40²) equal to 41²
8² + 40² = 64 + 1600 = 1664
41² = 1681
From the calculation, (8² + 40²) is not equal to 41²
We therefore conclude that triangle (a) is not a right-angled triangle.

Then we test (b)
Is (39² + 52²) equal to 65²
39² + 52² = 1521 + 2704 = 4225
65² = 4225
We therefore conclude that triangle (b) is a right-angled triangle.

Example 3.3

A frame is made from a wire in the shape of a trapezium as shown below. Calculate the length of the wire used.

We let the side that is not given be x.
The length of wire will be (9 + 8 + 15 + x) cm.
To find the length x, we deduce a right-angled triangle from the diagram.

x² = 6² + 8²
x² = 36 + 64
x² = 100
x = 10

The length of the wire is therefore 9 cm + 8 cm + 15 cm + 10 cm = 42 cm.

Build the Right Triangle

Adjust the values of a and b. Observe how the areas change and test the Pythagorean Relationship.

Side a (cm)
3

Side b (cm)
4

a² = 9

b² = 16

c² = 25

Exercise 3.1

Solve the following questions using the Pythagorean Relationship.

Question 1

(a) x = cm

(b) x = cm

Question 2

Determine which triangles are right-angled.

(a)

(b)

Question 3

A 3.9 m ladder is placed against a wall. The foot of the ladder is 1.5 m from the wall. How far up the wall does the ladder reach?

Answer: m

Question 4

A rectangle is 20 cm long and 15 cm wide. Find the length of the diagonal.

Diagonal = cm

Question 5

An airplane flies 480 km East and then 140 km North. How far is Wajir from Nairobi directly?

Distance = km

Question 6

AB = cm