The Baudhayana Pythagoras Theorem Class 8 Worksheet with Answers Maths Chapter 2

Each of our Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 2 The Baudhayana Pythagoras Theorem Worksheet with Answers focuses on conceptual clarity.

The Baudhayana Pythagoras Theorem Worksheet Class 8

Class 8 Maths The Baudhayana Pythagoras Theorem Worksheet

The Baudhayana Pythagoras Theorem Class 8 Ganita Prakash Worksheet

Doubling a Square

Riya: I want to make a square with double the area of this one. Should I just double the side?

Arjun: Walt! If you double the side, the area becomes four times bigger, not double. .

Riya: Oh! Then how can I double the area correctly?

Arjua: Take a paper cut-out of the square you want to double. Folding the square paper corner to corner and use its diagonal as the side of the new square.

Riya: Wow! That works. The new square’s area is exactly double the original.

Answer the following questions:
Question 1.
Why does doubling the sidelength of a square give four times the area instead of two? Explain with an example.
Answer:

Question 2.
Explain why the diagonal of a square can be used as the side of a new square to get double the area.
Answer:

Question 3.
If the original square has a sidelength of 6 cm, what will be the area of the new square when its diagonal is used as the side of the new square?
Answer:

Halving a Square

To halve a square means to find or construct another square whose area is one-half of the area of original square.

Question 4.
Draw a square of sidelength 10 cm. Show how to construct a square having half the area of the given square. Explain why the two parts are equal.
Answer:

Question 5.
In square ABCD, show that the triangles formed by diagonals are congruent.

Answer:

Hypotenuse of an Isosceles Right Triangle

An isosceles right triangle is a right-angled triangle in which the two sides containing the right angle are equal. In an isosceles right triangle, the hypotenuse is always s/2 times the length of one of the equal sides.

Question 6.
If each leg of an isosceles right triangle is 5 cm, what is the length of the hypotenuse?
Answer:
5√2 cm

Question 7.
Draw an isosceles right triangle with equal legs of 4 cm each. Show the hypotenuse and calculate its length.
Answer:
4√2 cm

Decimal Representation, of √2

√2 (square root of 2) is the number which, when multiplied by itself, gives 2, i.e., √2 × √2 = 2.
Question 8.
Why is √2 not a terminating decimal?

Question 9.
Find the lower and upper bounds of √2 till 5 decimal places.
Answer:
1.41421, 1.41422

Question 10.
The length of the two equal sides of an isosceles right triangle is given. Find the length of the hypotenuse. Find bounds on the length of the hypotenuse such that they have at least one digit after the decimal point.
(a) 5 units
(b) 7 units
(c) 11 units
Answer:
(a) 7.05 < 5√2 < 7.1
(b) 9.8 < 5√2 < 9.9
(c) 15.5 <1 1√2 < 15.6

Question 11.
The hypotenuse of an isosceles right triangle is 10√2 units. What are its other two sidelengths?
Answer:
10 units each

Combining Two Different Squares

When two squares of different side lengths are combined, the total area of the new shape is simply the sum of the areas of the two squares. The area of each square is found using the formula side x side. The way the squares are placed—side by side, touching, or arranged differently does not change the total area. Baudhayana’s Theorem on Right-angled triangles states that if a right-angled triangle has sidelengths a, b, and c, where c is the length of the hypotenuse, then a² + b² = c².

Question 12.
What is the combined areas of two squares with sides 4 cm and 3 cm?
Answer:
25 cm²

Question 13.
Draw two squares with sides 5 cm and 3 cm side by side. Find their total area. Then construct and shade one square whose area is equal to this combined area.
Answer:
34 cm²

Question 14.
If a right-angled triangle has shorter sides of lengths 1.5 cm and 2 cm, then what is the length of its hypotenuse? First draw the right-angled triangle with these sidelengths and measure the hypotenuse, then check your answer using Baudhayana’s Theorem.
Answer:
2.5 cm

Question 15.
If a right-angled triangle has a short side of length 7 cm and hypotenuse of length 25 cm, what is the length of the third side? Again, try drawing the triangle and measuring, and then check your answer using Baudhayana’s Theorem.
Answer:
24 cm

Question 16.
Find the missing sidelength (a, b are sides of a right-angled triangle, and c is hypotenuse):
(a) a = 20 cm, fa = 21 cm, find c.
(b) a = 9 cm, fa = 40 cm, find c.
(c) a = 11 cm, c = 61 cm, find fa.
(d) fa = 24 cm, c = 25 cm, find a.
Answer:
(a) c = 41 cm
(b) b = 60 cm
(c) a = 7 cm

Right-Triangles Having Integer Sidelengths

Question 17.
Show that the numbers (9, 12, 15) form the sidelength a right triangle.
Answer:

Question 18.
Verify using the Baudhayana-Pythagoras theorem that the triangle with sides 6 cm, 8 cm, and 10 cm is a right triangle.
Answer:

Question 19.
Identify which of the following Is a primitive Baudhayana triple: (3, 4, 5), (6, 8, 10), or (9, 12, 15).
Answer:
(3, 4, 5)

Question 20.
Generate three scaled versions of the primitive Baudhayana triple (5, 12, 13).
Answer:
(10, 24, 26), (15,36,39), (20, 48,52)

A Long-Standing Open Problem

Question 21.
State Fermats last theorem. Can you find a perfect cube that can be written as the sum of two perfect cubes?

Further Applications of The Baudhayana-Pythagoras Theorem
Baudhayana-Pythagoras Theorem is used to find missing sides of right-angled triangles, as well as in real-life situations such as measuring heights, distances, diagonals, or depths, etc.
Answer:
No

Question 22.
A tree casts a shadow 12 m long on the ground. A person measures the distance from the top of the tree to the tip of the shadow and finds it is 13 m. Find the height of the tree.
Answer:
5 m

Question 23.
Find the diagonal of a square with sidelength 7 cm.
Answer:
7 √2 cm

Question 24.
Find the missing sidelengths in the following right triangles:

Answer:
(a) 36
(b) 40
(c) 2√30
(d) √157

Question 25.
Find the sidelength of a rhombus whose diagonals are of length 6 units and 8 units.
Answer:
5 units

Question 26.
Construct a square whose area is equal to the difference of the areas of squares of sidelengths 7 units and 11 units.
Answer:

Worksheet On The Baudhayana Pythagoras Theorem Class 8

A. Choose the correct option.

1. The hypotenuse of an Isosceles right-angled triangle with legs 6 cm each is:
(a) 6 cm
(b) 6 cm
(c) 12 cm
(d) 3√2 cm
Answer:
(b) 6 cm

2. Doubling the side of a square increases its area by:
(a) 2 times
(b) 4 times
(c) Half
(d) Same
Answer:
(b) 4 times

3. A triangle has sides 2 m, 1.5 m, 2.5 m. This triangle is:
(a) Equilateral
(b) Right-angled
(c) Isosceles
(d) Scalene but not right-angled
Answer:
(b) Right-angled

4. The sum of areas of two squares with sides 6 cm and 7 cm is:
(a) 7 cm²
(b) 13 cm²
(c) 85 cm²
(d) 16 cm²
Answer:
(c) 85 cm²

5. The diagonal of a square of side 8 cm is:
(a) 16 cm
(b) 8√2 cm
(c) 64 cm
(d) \(\frac{8}{\sqrt{2}}\) cm
Answer:
(b) 8√2 cm

6. A triangle with sides (3k, 4k, 5k) for integer k is:
(a) Right-angled
(b) Equilateral
(c) Isosceles
(d) Not a triangle
Answer:
(a) Right-angled

7. Fermat’s Last Theorem states that for natural numbers x, y, z, and n > 2, the equation (xⁿ + y^a = zⁿ) has:
(a) Infinite solutions
(b) No solution in natural numbers
(c) One solution only
(d) Only prime number solutions
Answer:
(b) No solution in natural numbers

8. Halving the side of a square changes its area to:
(a) Half
(b) One-fourth
(c) Same
(d) Double
Answer:
(b) One-fourth

Directions. (For Q.9 – 10): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.

9. Assertion (A): The decimal representation of the square root of 2 is non-terminating.
Reason (R): Since √ 2 can be expressed as a fraction of two integers, it results in an infinite, non-terminating decimal.
Answer:
(c) Assertion (A) is true but Reason (R) is false.

10. Assertion (A): A scaled version of a Baudhayana triple (ka, kb, kc) is also a Baudhayana triple
Reason (R): If (a, b, c) is a Baudhayana triple, multiplying each term by a constant k results in another valid triple, as shown by the equation (ka)² + (kb)² = (kc)².
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

B. Fill in the Blanks.

1. The area of a square with side 8 cm is ______ cm².
Answer:
64

2. The hypotenuse of an isosceles right triangle is always ______ times one of the legs.
Answer:
√2

3. The Baudhayana triple (3, 4, 5) satisfies the equation ______.
Answer:
3 ² + 4² = 5²

4. Doubling a square’s area can be done by using the ______ of the original square as the side of the new square.
Answer:
diagonal

5. The Baudhayana theorem is also called the ______ theorem.
Answer:
Baudhayana-Pythagoras

C. State whether the following statements are true (T) or false (F).

1. Halving the side of a square always halves its area.
Answer:
false

2. The hypotenuse of an isosceles right triangle is V2 times the leg.
Answer:
True

3. (6, 8, 10) is a Baudhayana triple.
Answer:
True

4. Doubling the side of a square doubles its area.
Answer:
false

5. Combining squares changes their total area if arranged differently.
Answer:
false

D. Match the following.

Question 1.

Column A	Column B
1. Area of a square of sidelength 5 cm.	(a) 7 √2cm
2. A triangle with sidelengths (3, 4, 5).	(b) 25 cm2
3. Hypotenuse of an isosceles right triangle with equal leg of 7 cm.	(c) 65 cm2
4. Combined area of squares of sides 7 cm and 4 cm.	(d) Right-angled triangle

Answer:

Column A	Column B
1. Area of a square of sidelength 5 cm.	(b) 25 cm<sup>2</sup>
2. A triangle with sidelengths (3, 4, 5).	(d) Right-angled triangle
3. Hypotenuse of an isosceles right triangle with equal leg of 7 cm.	(a) 7 √2cm
4. Combined area of squares of sides 7 cm and 4 cm.	(c) 65 cm<sup>2</sup>

E. Solve the following.

Question 1.
Explain step by step how to double the area of a square using the diagonal.
Answer:

Question 2.
An Isosceles right triangle has equal legs 8 cm each. Find the hypotenuse and show all calculations.
Answer:
8√2 cm

Question 3.
A lotus plant grows in a lake with its stem standing vertically. The tip of the lotus is 2 units above the surface of the water. When the lotus is bent by the wind, its tip touches the water at a point 5 units away from the place where the stem emerges from the water. Find the depth of the lake.
Answer:
5. 25  units

Question 4.
Find the missing side of a right triangle if the hypotenuse is 13 cm and one side is 5 cm.
Answer:
12 cm

Question 5.
A triangular ramp to be made for a tree house. The base and height of the ramp are equal in length. Calculate the length of the ramp in terms of its height.
Answer:
√2 × height