Maharashtra Board Class 7 Maths part 2 Chapter 13 Pythagoras Theorem PDF Download

Pythagoras' Theorem

Let's Recall

Right-Angled Triangle

We know that a triangle with one right angle is called a right-angled triangle and the side opposite to the right angle is called the hypotenuse.

Write the name of the hypotenuse of each of the right-angled triangles shown below.

The hypotenuse of Triangle ABC is AC.

The hypotenuse of Triangle LMN is LN.

The hypotenuse of Triangle XYZ is XZ.

Pythagoras' Theorem

Pythagoras was a great Greek mathematician of the 6th century BCE. He made important contributions to mathematics. His method of teaching mathematics was very popular. He trained several mathematicians.

People of many countries had long known of a certain principle related to the right-angled triangle. It is also given in the book called Shulvasutra of ancient India. As Pythagoras was the first to prove the theorem, it is named after him.

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

Activity

Draw right-angled triangles given the hypotenuse and one side as shown in the rough figures below. Measure the third side. Verify Pythagoras' theorem.

(i) Triangle ABC with hypotenuse AC = 5 and base BC = 3

(ii) Triangle PQR with hypotenuse PR = 10 and side QR = 6

(iii) Triangle XYZ with hypotenuse XZ = 17 and base XY = 15

Teacher's Note

Pythagoras' theorem helps us find missing sides in right-angled triangles. For example, if a ladder is leaning against a wall in India, we can use this theorem to find how far the ladder is from the wall.

Exam Trick

Remember: The longest side is always the hypotenuse. Square it and set it equal to the sum of squares of the other two sides. Just like \(5^2 = 3^2 + 4^2\), which is \(25 = 9 + 16\).

Points to Remember

A right-angled triangle has one angle that is 90 degrees.
The hypotenuse is the side opposite to the right angle.
Pythagoras' theorem says: (hypotenuse)² = (base)² + (height)².
This theorem is very useful in finding missing sides of triangles.

Let's Learn

With reference to the figure alongside, the theorem of Pythagoras can be written as follows:

In triangle ABC, if angle B is a right angle, then \([l(AC)]^2 = [l(AB)]^2 + [l(BC)]^2\)

Generally, in a right-angled triangle, one of the sides forming the right angle is taken as the base and the other as the height.

Then, the theorem can be stated as:

\[(\text{hypotenuse})^2 = (\text{base})^2 + (\text{height})^2\]

Activity: Verify Pythagoras' Theorem

From a cardsheet, cut out eight identical right-angled triangles. Let us say the length of the hypotenuse of these triangles is 'a' units and sides forming the right angle are 'b' and 'c' units. Note that the area of this triangle is \(\frac{bc}{2}\).

Next, on another cardsheet, use a pencil to draw two squares ABCD and PQRS each of side (b + c) units. Now place 4 of the triangle cut-outs in the square ABCD and the remaining 4 in the square PQRS as shown in the figures below. Mark by lines drawn across them, the parts of the squares covered by the triangles.

Observe the figures. In figure (i) we can see a square of side a units in the uncovered portion of square ABCD. In figure (ii) we see a square of side b and another of side c in the uncovered portion of the square PQRS.

In figure (i), area of square ABCD = \(a^2 + 4 \times\) area of right-angled triangle

\[= a^2 + 4 \times \frac{1}{2}bc = a^2 + 2bc\]

Teacher's Note

This activity shows us that Pythagoras' theorem is true by using squares and triangles. In India, this same idea was used in ancient math books like the Shulvasutra.

Exam Trick

When you see two squares placed with triangles, count the uncovered area. One big square equals two smaller squares plus triangles. This helps you see why \(a^2 = b^2 + c^2\).

Points to Remember

The area of a right-angled triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\).
When triangles are placed in squares, the uncovered area shows us the relationship.
Two small squares' areas together equal one big square's area.
This visual proof helps us understand why the theorem is true.

In figure (ii), area of square PQRS = \(b^2 + c^2 + 4 \times\) area of right-angled triangle

\[= b^2 + c^2 + 4 \times \frac{1}{2}bc = b^2 + c^2 + 2bc\]

Area of square ABCD = Area of square PQRS

\[\therefore a^2 + 2bc = b^2 + c^2 + 2bc\]

\[\therefore a^2 = b^2 + c^2\]

Let's Discuss

Without using a protractor, can you verify that every angle of the vacant quadrilateral in figure (i) is a right angle?

Activity

On a sheet of card paper, draw a right-angled triangle of sides 3 cm, 4 cm and 5 cm. Construct a square on each of the sides. Find the area of each of the squares and verify Pythagoras' theorem.

Teacher's Note

Making squares on a triangle's sides helps students see the theorem. If you draw a 3-4-5 triangle on paper and make squares, you will see that the big square has area 25, which equals 9 + 16.

Exam Trick

The 3-4-5 triangle is the easiest to remember. \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\). Use this as your first example in exams.

Points to Remember

A 3-4-5 triangle is the simplest right-angled triangle.
The square on side 3 has area 9 square cm.
The square on side 4 has area 16 square cm.
The square on side 5 has area 25 square cm.
We can verify that 9 + 16 = 25.

Given two sides of a right-angled triangle, you can find the third side, using Pythagoras' theorem.

Example

In triangle ABC, angle C = 90°, l(AC) = 5 cm and l(BC) = 12 cm. What is the length of seg (AB)?

Solution

In the right-angled triangle ABC, angle C = 90°.

Hence, side AB is the hypotenuse.

According to Pythagoras' theorem, \([l(AB)]^2 = [l(AC)]^2 + [l(BC)]^2\)

\[= 5^2 + 12^2 = 25 + 144 = 169\]

\[[l(AB)]^2 = 13^2\]

\[l(AB) = 13\]

\[\therefore \text{Length of seg AB} = 13 \text{ cm}\]

Teacher's Note

This example shows how to find a missing side. If you know any two sides of a right triangle, you can always find the third one using the formula.

Exam Trick

Always identify which side is the hypotenuse first (the longest side). Then apply the formula. Write step-by-step to get full marks.

Points to Remember

The hypotenuse is always opposite the right angle.
Square both known sides and add them or subtract them carefully.
Take the square root at the end to find the missing side.
Check your answer by verifying the theorem again.

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