## AP 9th Class Maths Bits 10th Lesson Heron’s Formula

Multiple Choice Questions (MCQs) :

Question 1.

Area of what type triangle can be solved using Heron’s formula ?

A) Right triangle

B) Isosceles triangle

C) Equilateral triangle

D) Any triangle

(OR)

Heron has derived the formula for the area of _________

A) equilateral triangle

B) scalene triangle

C) isosceles triangle

D) any triangle

Answer:

D) any triangle

Question 2.

Which of the following formulas can be used to find the area of a triangle using Heron’s formula ?

A) A = \(\frac{1}{2}\)bh

B) A = bh

C) A = \(\frac{1}{2}\)ab sin C

D) A = \(\sqrt{(s(s-a)(s-b)(s-c)}\)

Answer:

D) A = \(\sqrt{(s(s-a)(s-b)(s-c)}\)

Question 3.

If a triangle has sides of lengths 5 cm, 12 cm and 13 cm, what is its area using Heron’s formula ?

A) 20 cm^{2}

B) 30 cm^{2}

C) 24 cm^{2}

D) 60 cm^{2}

Answer:

B) 30 cm^{2}

Question 4.

Heron’s formula can be used to find the area of a triangle if you know the lengths of ________

A) All three sides

B) Two sides and an angle

C) Two angles and a side

D) One side and two angles

Answer:

A) All three sides

Question 5.

In a triangle, if the semi perimeter is s and the lengths of the sides are a, b and c, then what is the value of s in terms of a, b and c ?

A) s = a + b

B) s = a + b + c

C) s = (a + b + c)/2

D) s = (a + b – c)/2

Answer:

C) s = (a + b + c)/2

Question 6.

What is the maximum area of a triangle with sides of lengths 6 cm, 8 cm and 10 cm ?

A) 12 cm^{2}

B) 24 cm^{2}

C) 30 cm^{2}

D) 48 cm^{2}

Answer:

B) 24 cm^{2}

Question 7.

If a triangle has sides of lengths 7 cm, 8 cm and 9 cm and the altitude from the longest side is 6 cm, what is the area of the triangle using Heron’s formula ?

A) 20 cm^{2}

B) 12√5 cm^{2}

C) 30 cm^{2}

D) 36 cm^{2}

Answer:

B) 12√5 cm^{2}

Question 8.

Heron’s formula is named after

A) Euclid

B) Archimedes

C) Pythagoras

D) Hero of Alexandria

Answer:

D) Hero of Alexandria

Question 9.

A triangle has sides of length 12, 13 and 14. What,is the area of the triangle using Heron’s formula ?

A) 72

B) 72.31

C) 90

D) 96

Answer:

B) 72.31

Question 10.

The sides of a triangle are 25, 25 and 30. What is the area of the triangle using Heron’s formula?

A) 300

B) 192.3

C) 196.9

D) 200.2

Answer:

A) 300

Question 11.

The sides of a triangle are 7, 9 and 13. What is the area of the triangle using Heron’s formula ?

A) 12.97

B) 15.25

C) 18.81

D) 29.95

Answer:

D) 29.95

Question 12.

The sides of a triangle are in the ratio 3 : 8 : 9 and its perimeter, is 132. What is the area of the triangle using Heron’s formula?

A) 716

B) 716.4

C) 716.3

D) 716.5

Answer:

B) 716.4

Question 13.

The sides of a right triangle are 5, 12, and 13. What is the area of the triangle using Heron’s formula?

A) 30

B) 24

C) 36

D) 40

Answer:

A) 30

Question 14.

The perimeter of an isosceles triangle is 36 and its two equal sides are 13 each. What is the area of the triangle using Heron’s formula?

A) 84

B) 60

C) 96

D) 102

Answer:

B) 60

Question 15.

The perimeter of a right triangle is 30 and its hypotenuse is 13. What is the area of the triangle using Heron’s formula ?

A) 30

B) 60

C) 84

D) 156

Answer:

A) 30

Question 16.

What is Heron’s formula used for ?

A) Finding the area of a circle

B) Finding the area of a triangle

C) Finding the perimeter of a quadrilateral

D) Finding the volume of a sphere

Answer:

B) Finding the area of a triangle

Question 17.

Which of the following is the correct formula for Heron’s formula?

A) A = bh

B) A = \(\frac{1}{2}\)bh

C) A = \(\sqrt{s(s-a)(s-b)(s-c)}\)

D) A = lb

Answer:

C) A = \(\sqrt{s(s-a)(s-b)(s-c)}\)

Question 18.

What is the semi-perimeter of a triangle?

A) The sum of the lengths of the sides of a triangle

B) Half of the sum of the lengths of the sides of a triangle

C) Half of the difference between the lengths of the sides of a triangle

D) The product of the lengths of the sides of a triangle

Answer:

B) Half of the sum of the lengths of the sides of a triangle

Question 19.

What is the maximum number of sides a triangle can have?

A) 2

B) 3

C) 4

D) 5

Answer:

B) 3

Question 20.

If a triangle has sides of length 3, 4 and 5, what is its area using Heron’s formula ?

A) 6

B) 8

C) 10

D) 12

Answer:

A) 6

Question 21.

What is the area of an equilateral triangle with sides of length 6 using Heron’s formula ?

A) 9.75

B) 15.59

C) 18

D) 20.78

Answer:

A) 9.75

Question 22.

What is the area of an isosceles triangle with sides of length 5, 5 and 6 using Heron’s formula ?

A) 6

B) 7.8

C) 8.64

D) 9

Answer:

C) 8.64

Question 23.

Which of the following is not required to use Heron’s formula ?

A) The length of all three sides of the triangle

B) The perimeter of the triangle

C) The semi-perimeter of the triangle

D) The height of the triangle

Answer:

D) The height of the triangle

Question 24.

What is the area of a right triangle with sides of length 3 and 4 using Heron’s formula?

A) 3

B) 4

C) 5

D) 6

Answer:

C) 5

Question 25.

Area of a triangle =

A) \(\frac{1}{2}\) × Base × Height

B) Base × Height

C) \(\frac{1}{3}\) × Base × Height

D) \(\frac{1}{4}\) × Base × Height

Answer:

A) \(\frac{1}{2}\) × Base × Height

Question 26.

The area of ΔABC in which AB = BC = 4 cm and ∠B = 90° is

A) 16 cm^{2}

B) 8 cm^{2}

C) 4 cm^{2}

D) 12cm^{2}

Answer:

B) 8 cm^{2}

Question 27.

Area of a triangle having base 6 cm and altitude 8 cm is

A) 48 cm^{2}

B) 24 cm^{2}

C) 64 cm^{2}

D) 36 cm^{2}

Answer:

B) 24 cm^{2}

Question 28.

Area of a triangle is 60 cm^{2}. Its base is 15 cm. Its altitude is

A) 30 cm

B) 4 cm

C) 8 cm

D) 10 cm

Answer:

C) 8 cm

Question 29.

The area of a right triangle is 36 cm^{2} and its base is 9 cm. Find the length of the perpendicular

A) 8 cm

B) 4 cm

C) 16 cm

D) 32 cm

Answer:

A) 8 cm

Question 30.

The side of an isosceles right angle triangle of hypotenuse 5√2 cm is

A) 10 cm

B) 8 cm

C) 5 cm

D) 3√2 cm

Answer:

C) 5 cm

Question 31.

The base of a right triangle is 15 cm and its hypotenuse is 25 cm. Then its area is

A) 187.5 cm^{2}

B) 375 cm^{2}

C) 150 cm^{2}

D) 300 cm^{2}

Answer:

C) 150 cm^{2}

Question 32.

The side of an equilateral triangle is 6 cm. The area of the triangle is

A) 6√3 cm^{2}

B) 9√3 cm^{2}

C) 16√3 cm^{2}

D) 3√3 cm^{2}

Answer:

B) 9√3 cm^{2}

Question 33.

Side of an equilateral triangle is 4 cm. Its area is

A) 4√3 cm^{2}

B) \(\frac{\sqrt{3}}{4}\) cm^{2}

C) √3 cm^{2}

D) 2√3 cm^{2}

Answer:

A) 4√3 cm^{2}

Question 34.

Find the length of the side of an equi-lateral triangle whose area is 9√3 m^{2}

A) 1 cm

B) 2 cm

C) 3 cm

D) 6 cm

Answer:

D) 6 cm

Question 35.

The area of an equilateral triangle is 16√3 m^{2}. Its perimeter (in metres) is

A) 12

B) 48

C) 24

D) 306

Answer:

C) 24

Question 36.

The perimeter of an equilateral triangle is 60 m. Its area is

A) 10√3m^{2}

B) 100√3m^{2}

C) 15√3m^{2}

D) 20√3 m^{2}

Answer:

B) 100√3m^{2}

Question 37.

If the length of median of an equilateral triangle be x cm, then its area is

A) x^{2}

B) \(\frac{\sqrt{3}}{2}\)x^{2}

C) \(\frac{\mathrm{x}^2}{\sqrt{3}}\)

D) \(\frac{x^2}{2}\)

Answer:

C) \(\frac{\mathrm{x}^2}{\sqrt{3}}\)

Question 38.

The sides of a triangle are 7 cm, 24 cm and 25 cm. Its area is

A) 168 cm^{2}

B) 84 cm^{2}

C) 87.5 cm^{2}

D) 300 cm^{2}

Answer:

B) 84 cm^{2}

Question 39.

Area of an equilateral triangle of side ‘a’ is

A) \(\frac{\sqrt{3}}{4}\)a^{2}

B) \(\frac{\mathrm{a} \sqrt{3}}{2}\)

C) √3a^{2}

D) a√3

Answer:

A) \(\frac{\sqrt{3}}{4}\)a^{2}

Question 40.

Area of an equilateral triangle of side ‘a’ units can be calculated by using the formula

A) \(\sqrt{s^2(s-a)^2}\)

B) (s – a)\(\sqrt{s^2(s-a)}\)

C) \(\sqrt{\mathrm{s}(\mathrm{s}-\mathrm{a})^2}\)

D) (s – a)\(\sqrt{s(s-a)}\)

Answer:

D) (s – a)\(\sqrt{s(s-a)}\)

Question 41.

Find the perimeter of the triangle whose sides are 17 cm, 33 cm and 20 cm

A) 70 cm

B) 50 cm

C) 53 cm

D) 37 cm

Answer:

A) 70 cm

Question 42.

The semi-perimeter of a triangle having the length of its sides as 20 cm, 15 cm and 9 cm is

A) 44 cm

B) 21 cm

C) 22 cm

D) None

Answer:

C) 22 cm

Question 43.

The perimeter of a triangular plot is 16 m. If the measures of its two sides are 5 m and 6 m, then find the third side

A) 2 m

B) 3 m

C) 5 m

D) 4 m

Answer:

C) 5 m

Question 44.

Two sides of a triangle are 13 cm and 14 cm and its semi-perimeter is 18 cm. Then third side of the triangle is

A) 12 cm

B) 11 cm

C) 10 cm

D) 9 cm

Answer:

D) 9 cm

Question 45.

The area of a triangle whose sides are 13 cm, 14 cm and 15 cm is

A) 42 cm^{2}

B) 86 cm^{2}

C) 84 cm^{2}

D) 100 cm^{2}

Answer:

C) 84 cm^{2}

Question 46.

Two equal rides of an isosceles triangle are 13 cm each and its perimeter is 36 cm. Find the area of the triangle.

A) 20 cm^{2}

B) 30 cm^{2}

C) 40 cm^{2}

D) 60 cm^{2}

Answer:

D) 60 cm^{2}

Question 47.

Find the area of an isosceles triangle whose equal sides are 6 cm each and the third side is 8 cm.

A) 8√5 cm^{2}

B) 5√8 cm^{2}

C) 3\(\sqrt{55}\) cm^{2}

D) 3√8 cm^{2}

Answer:

A) 8√5 cm^{2}

Question 48.

The perimeter of a triangle is 36 cm and its sides are in the ratio a : b : c = 3 : 4 : 5, then a, b, c are respectively.

A) 9 cm, 15 cm, 12 cm

B) 15 cm, 12 cm, 9 cm

C) 12 cm, 9 cm, 15 cm

D) 9 cm, 12 cm, 16 cm

Answer:

D) 9 cm, 12 cm, 16 cm

Question 49.

The sides of a triangular field are in the ratio 3 : 4 : 5. The perimeter of the f triangular field is 144 m. Find the longest side of the field.

A) 15 m

B) 30 m

C) 60 m

D) 90 m

Answer:

C) 60 m

Question 50.

Area of a quadrilateral =

A) \(\frac{1}{2}\) × a diagonal × sum of the perpendiculars on the diagonal

B) a diagonal × sum of the perpendiculars on the diagonal

C) \(\frac{1}{3}\) × a diagonal × sum of the perpendiculars on the diagonal

D) \(\frac{1}{4}\) × a diagonal × sum of the perpendiculars on the diagonal

Answer:

A) \(\frac{1}{2}\) × a diagonal × sum of the perpendiculars on the diagonal

Question 51.

Area of a trapezium =

A) \(\frac{1}{2}\) × sum of parallel sides × distance between the parallel sides

B) sum of parallel sides × distance between the parallel sides

C) \(\frac{1}{3}\) × sum of parallel sides × distance between the parallel sides

D) None of these

Answer:

A) \(\frac{1}{2}\) × sum of parallel sides × distance between the parallel sides

Question 52.

1 hectare =

A) 10 m^{2}

B) 100 m^{2}

C) 1000 m^{2}

D) 10000 m^{2}

Answer:

D) 10000 m^{2}

Question 53.

1 acre =

A) 10 m^{2}

B) 100 m^{2}

C) 1000 m^{2}

D) 10000 m^{2}

Answer:

B) 100 m^{2}

Question 54.

Find the area of a right angled triangle, if the radius of the semi-circle is 3 cm and altitude drawn to the hypotenuse is 2 cm

A) 4 cm^{2}

B) 6 cm^{2}

C) 8 cm^{2}

D) 12 cm^{2}

Answer:

B) 6 cm^{2}

Question 55.

The side of a square is 5 cm. Its perimeter is

A) 5 cm

B) 20 cm

C) 25 cm

D) 10 cm

Answer:

B) 20 cm

Question 56.

Find the area of a quadrilateral whose one diagonal is 8 cm and the sum of perpendiculars from vertices is 10 cm

A) 20 cm^{2}

B) 40 cm^{2}

C) 80 cm^{2}

D) 160 cm^{2}

Answer:

B) 40 cm^{2}

Question 57.

The diagonals of a rhombus are 10 cm and 8 cm. Its area is

A) 80 cm^{2}

B) 40 cm^{2}

C) 9 cm^{2}

D) 36 cm^{2}

Answer:

B) 40 cm^{2}

Question 58.

The area of a rhombus is 96 cm^{2}. If one of its diagonals is 16 cm, then the length of its sides is

A) 12 cm

B) 10 cm

C) 8 cm

D) 6 cm

Answer:

B) 10 cm

Question 59.

The perimeter of a rhombus is 20 cm. If one of its diagonals is 6 cm, then its area is

A) 28 cm^{2}

B) 36 cm^{2}

C) 24 cm^{2}

D) 20 cm^{2}

Answer:

C) 24 cm^{2}

Question 60.

The parallel sides of a trapezium are 45.8 cm and 81.2 cm and the distance, between them is 22 cm. Find the area of the trapezium.

A) 1397 cm^{2}

B) 1937 cm^{2}

C) 3197 cm^{2}

D) 139.7 cm^{2}

Answer:

A) 1397 cm^{2}

Question 61.

A regular hexagon has a side 8 cm. Find its area.

A) 8√3 cm^{2}

B) 96√3 cm^{2}

C) 4√3 cm^{2}

D) 12√3 cm^{2}

Answer:

B) 96√3 cm^{2}

Question 62.

In the Heron’s formula which of the following is correct ?

A) S is perimeter of triangle

B) S is semi-perimenter of triangle

C) 2S is semi-perimeter of triangle S

D) \(\frac{\mathrm{S}}{2}\) is perimeter of triangle

Answer:

B) S is semi-perimenter of triangle

Assertion and Reason type questions :

Question 1.

Assertion : Heron’s formula can be used to find the area of any triangle.

Reason : Heron’s formula uses1 the , length of the sides of the triangle to calculate its area.

Answer:

Assertion is true.

Reason is true and explains the assertion.

Explanation : Heron’s formula can be used to find the area of any triangle regardless of its shape or size. This is because the formula uses the length oi the suies of the triangle to calculate its area, which means that as long as you know the lengths of the sides, you can use the formula to find the area. Therefore, the assertion is true and the reason is also true and explains why Heron’s formula can be used to find the area of any triangle.

Question 2.

Assertion : Heron’s formula is named after the ancient Greek mathematician Euclid.

Reason : Euclid was the first person to discover the formula for finding the area of a triangle.

Answer:

Assertion is false.

Reason is false.

Explanation : The assertion is false because Heron’s formula is actually named after the ancient Greek mathematician Heron of Alexandria, who discovered the formula. The reason is also false because while Euclid did write extensively on geometry, he did not discover Heron’s formula for finding the area of a triangle.

Question 3.

Assertion : Heron is formula can be used to find the area of a triangle even if only two sides and an angle opposite one of the sides are known.

Reason : Heron’s formula involves the use of the Pythagorean theorem to find the area of a triangle.

Answer:

Assertion is false.

Reason is false.

Explanation ; The assertion is false because Heron’s formula requires the lengths of all three sides of a triangle to be known, not just two sides and an opposite angle. The reason is also false because while the Pythagorean theorem is used in some formulas for finding the area of a triangle, it is not used in Heron’s formula.

Question 4.

Assertion : Heron’s formula is more accurate for finding the area of a triangle than the formula A = 1/2bh.

Reason : Heron’s formula takes into account the lengths of all three sides of the triangle, while A = 1/2bh only takes into account the base and height.

Answer:

Assertion is true.

Reason is true and explains the assertion.

Explanation : Heron’s formula is more accurate for finding the area of a triangle than the formula A = \(\frac{1}{2}\) bh because it takes into account the lengths of all three sides of the triangle, which means that it can hp used to find the area of any triangle, regardless of its shape or size. In contrast, A = \(\frac{1}{2}\) bh only takes into account the base and height of a triangle, which means that it can only be used to find the area of certain types of triangles, such as right triangles. Therefore, the assertion is true and the reason is , also true and explains why Heron’s formula is more accurate.

Question 5.

Assertion : Heron’s formula can be Used to find the area of a triangle with sides of length 0.

Reason : Heron’s formula involves dividing by zero in certain cases, which means that it can be used to find the area of any triangle, including those with sides of length 0.

Answer:

Assertion is false.

Reason is false.

Explanation : The assertion is false because Heron’s formula cannot be used to find the area of a triangle with sides of length 0. This is because the formula involves taking the square root of a number and the square root of a negative number (which results when one or more sides are 0) is undefined. The reason is also false because Heron’s formula does not involve dividing by zero in any case, but rather involves taking the square root of a value that is always greater than or equal to zero.

Fill in the blanks :

1. Heron’s formula gives the area of a triangle in terms of the lengths of its ___________ .

Answer:

sides

2. The semiperimeter of a triangle is defined as ___________ of the lengths of its sides.

Answer:

half the sum

3. The area of a triangle with side lengths 5 cm, 12 cm and 13 cm is _________ cm^{2}.

Answer:

30

4. Heron’s formula can be used to find the area of a triangle when the lengths of ___________ are given.

Answer:

all three sides

5. The formula for the semiperimeter of a triangle with sides a, b and c is s = ___________ .

Answer:

(a + b + c) / 2

6. A triangle with side lengths 7 cm, 8 cm and 9 cm has an area of ___________ cm^{2}.

Answer:

26.83 (rounded to two decimal places)

7. If a triangle has sides of length 6 cm, 8 cm and 10 cm, then its area is __________ cm^{2}.

Answer:

24

8. Heron’s formula is named after _________ a Greek mathematician and engineer who lived in the first century AD.

Answer:

Hero of Alexandria

9. The area of a triangle with sides of length 12 cm, 16 cm and 20 cm can be found using Heron’s formula to be _________ cm^{2}.

Answer:

96

10. The area of an equilateral triangle with side length 10 cm can be found using Heron’s formula to be __________ cm^{2}.

Answer:

43.30 (rounded to two decimal places)

11. The area of a triangle with sides of length 5, 7 and 8 can be found using Heron’s formula to be ___________ .

Answer:

10√3 sq units

12. If one side of a right triangle are in 5 cm and the length of the hypotenuse is 13cm, then the area of the triangle can be found using Heron’s formula to be ___________ .

Answer:

30

13. If the sides of a triangle are in the ratio 3:4:5 and the semiperimeter of the tri-angle is 30 cm .then its area is ____________ cm^{2}.

Answer:

150

14. If the length of two sides of a triangle are 15 cm and 8 cm and the angle between them is 90 degrees, then the area of the triangle can be found using Heron’s formula to be __________ .

Answer:

60 cm^{2}

15. The sides of a triangle are in the ratio 7 : 9 : 11 and the perimeter of the triangle is 54 cm. Then the area of the triangle can be found using Heron’s formula to be ___________ .

Answer:

125.68 cm^{2}

16. If the perimeter of an isosceles triangle is 26 cm and the length of the inequal sides is 10 cm, then the area of the triangle can be found using Heron’s for-mula to be ___________ .

Answer:

31.22 cm^{2}

17. If the area of a triangle is 12√3 cm^{2} and the semiperimeter is 12 cm and if two lengths of the sides are 6 and 10 cm, then the third side of the triangle will be ___________ .

Answer:

9 cm

18. If the area of a triangle is 24 cm^{2} and two sides of the triangle are 6 cm and 8 cm then the third side of the triangle can have lengths ___________ .

Answer:

10 cm

19. If the lengths of two sides of a triangle are 5 cm and 7 cm and the area of the triangle is 12 cm2, then the possible lengths of the third side of the triangle are ___________ .

Answer:

11.17