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The Baudhayana-Pythagoras Theorem - Doubling a Square

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Baudhayana-Pythagoras Theorem 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. In ancient Indian mathematics, Baudhayana described this as the 'diagonal of a rectangle producing as much area as the two sides produce together'.

Right-angled triangle showing sides a, b and diagonal d.
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Doubling a Square: If we take a square with side ss, its area is s2s^2. When we construct a new square using the diagonal of the original square as its side, the area of this new square is exactly double the area of the original square (2s22s^2).

Construction of a larger square (red) on the diagonal of a smaller square.
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Mathematical relationship: For a square of side ss and diagonal dd, the area of the square on the diagonal is d2=s2+s2=2s2d^2 = s^2 + s^2 = 2s^2. This provides a geometric method to find a square with twice the area of a given square.

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The length of the diagonal dd can be calculated as d=s2d = s\sqrt{2}. This represents the ratio of the diagonal of a square to its side.

📐Formulae

Area of a square=s2Area\ of\ a\ square = s^2

d2=s2+s2d^2 = s^2 + s^2

d2=2s2d^2 = 2s^2

d=s2d = s\sqrt{2}

💡Examples

Problem 1:

If a square has a side of 6 cm6\text{ cm}, find the area of the square formed by using its diagonal as a side.

Solution:

Given side s=6 cms = 6\text{ cm}. Area of original square = s2=62=36 cm2s^2 = 6^2 = 36\text{ cm}^2. According to the Baudhayana-Pythagoras Theorem, the square of the diagonal d2d^2 is: d2=2s2d^2 = 2s^2 d2=2×(6)2d^2 = 2 \times (6)^2 d2=2×36=72 cm2d^2 = 2 \times 36 = 72\text{ cm}^2.

Explanation:

The area of the square formed by the diagonal is exactly twice the area of the original square. Since the original area is 36 cm236\text{ cm}^2, the new area is 72 cm272\text{ cm}^2.

Problem 2:

Verify the doubling of a square with side 10 units10\text{ units} and calculate the increase in area using vertical subtraction.

Solution:

Original Area A1=102=100 sq unitsA_1 = 10^2 = 100\text{ sq units}. New Area (Doubled) A2=2×100=200 sq unitsA_2 = 2 \times 100 = 200\text{ sq units}. Difference in area: 200−100100\begin{array}{r} 200 \\ - 100 \\ \hline 100 \end{array} The increase in area is 100 sq units100\text{ sq units}.

Explanation:

By using the diagonal as the side of a new square, the area is doubled. Subtracting the original area from the new area shows that the area added is equal to the original area.

Problem 3:

A square has a side length of 7 cm7\text{ cm}. Find the area of a second square whose side is equal to the diagonal of the first square.

Square with side 7cm and diagonal d.

Solution:

Area of original square=72=49 cm2Area\ of\ original\ square = 7^2 = 49\text{ cm}^2 Area of new square=2×49=98 cm2Area\ of\ new\ square = 2 \times 49 = 98\text{ cm}^2 Alternatively, using d2=s2+s2:\text{Alternatively, using } d^2 = s^2 + s^2: d2=72+72=49+49=98 cm2d^2 = 7^2 + 7^2 = 49 + 49 = 98\text{ cm}^2

Explanation:

According to the property of doubling a square, the area of the square built on the diagonal is twice the area of the original square.

Problem 4:

A square garden has an area of 50 m250\text{ m}^2. If a path is constructed along its diagonal to form a larger square, find the area of this larger square and the difference in area between the two squares.

Blue square constructed on the diagonal of a garden square.

Solution:

Original Area=50 m2\text{Original Area} = 50\text{ m}^2 New Area=2×50=100 m2\text{New Area} = 2 \times 50 = 100\text{ m}^2 Difference in Area:\text{Difference in Area:} 100−5050\begin{array}{r} 100 \\ - 50 \\ \hline 50 \end{array} The difference is 50 m2.\text{The difference is } 50\text{ m}^2.

Explanation:

Since the larger square is formed using the diagonal of the smaller square, its area is double the smaller square's area. The difference is simply the area of the original square.