R.d Sharma 2022 Mcqs Solutions for Class 9 Maths Chapter 17 Heron' S Formula are provided here with simple step-by-step explanations. These solutions for Heron' S Formula are extremely popular among class 9 students for Maths Heron' S Formula Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the R.d Sharma 2022 Mcqs Book of class 9 Maths Chapter 17 are provided here for you for free. You will also love the ad-free experience on Meritnation’s R.d Sharma 2022 Mcqs Solutions. All R.d Sharma 2022 Mcqs Solutions for class 9 Maths are prepared by experts and are 100% accurate.
Page No 180:
Question 1:
Mark the correct alternative in each of the following:
The sides of a triangle are 16 cm, 30 cm, 34 cm. Its area is
(a) 225 cm2
(b) 240 cm2
(c) cm2
(d) 450 cm2
Answer:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangle say A, having sides 16 cm, 30 cm and 34 cm is given by
a = 16 cm ; b = 30 cm ; c = 34 cm
Therefore the area of the triangle is
Hence, the correct option is (b).
Page No 180:
Question 2:
The base of an isosceles right triangle is 30 cm. Its area is
(a) 225 cm2
(b) 225 cm2
(c) 225 cm2
(d) 450 cm2
Answer:
Hence, the correct option is (d).
Page No 180:
Question 3:
The sides of a triangle are 7 cm, 9 cm and 14 cm. Its area is
(a)
(b)
(c)
(d)
Answer:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangle having sides 7 cm, 9 cm and 14 cm is given by
a = 7 cm ; b = 9 cm ; c = 14 cm
Therefore the answer is (a).
Page No 180:
Question 4:
The sides of a triangular field are 325 m, 300 m and 125 m. Its area is
(a) 18750 m2
(b) 37500 m2
(c) 97500 m2
(d) 48750 m2
Answer:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangular field, say A having sides 325 m, 300 m and 125 m is given by
a = 325 m ; b = 300 m ; c = 125 m
Therefore, the correct answer is (a).
Page No 180:
Question 5:
The sides of a triangle are 50 cm, 78 cm and 112 cm. The smallest altitude is
(a) 20 cm
(b) 30 cm
(c) 40 cm
(d) 50 cm
Answer:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangle, say A having sides 50 cm, 78 cm and 112 cm is given by
The area of a triangle, having p as the altitude will be,
Where, A = 1680
We have to find the smallest altitude, so will substitute the value of the base AC with the length of each side one by one and find the smallest altitude distance i.e. p
Case 1
Case 2
Case 3
Therefore, the answer is (b).
Page No 180:
Question 6:
The sides of a triangle are 11 m, 60 m and 61 m. The altitude to the smallest side is
(a) 11 m
(b) 66 m
(c) 50 m
(d) 60 m
Answer:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
We need to find the altitude to the smallest side
Therefore the area of a triangle having sides 11 m, 60 m and 61 m is given by
a = 11 m ; b = 60 m ; c = 61 m
The area of a triangle having base AC and height p is given by
We have to find the height p corresponding to the smallest side of the triangle. Here smallest side is 11 m
AC = 11 m
Therefore, the answer is (d).
Page No 180:
Question 7:
The sides of a triangle are 11 cm, 15 cm and 16 cm. The altitude to the largest side is
(a)
(b)
(c)
(d) 30 cm
Answer:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
We need to find the altitude corresponding to the longest side
Therefore the area of a triangle having sides 11 cm, 15 cm and 16 cm is given by
a = 11 m ; b = 15 cm ; c = 16 cm
The area of a triangle having base AC and height p is given by
We have to find the height p corresponding to the longest side of the triangle.Here longest side is 16 cm, that is AC=16 cm
Therefore, the answer is (c).
Page No 180:
Question 8:
The base and hypotenuse of a right triangle are respectively 5 cm and 13 cm long. Its area is
(a) 25 cm2
(b) 28 cm2
(c) 30 cm2
(d) 40 cm2
Answer:
In right angled triangle ABC having base 5 cm and hypotenuse 13 cm we are asked to find its area
Using Pythagorean Theorem
Where, AB = hypotenuse = 13 cm, AC = Base = 5 cm, BC = Height
Area of a triangle, say A having base 5 cm and altitude 12 cm is given by
Where, Base = 5 cm; Height = 12 cm
Therefore, the answer is (c).
Page No 180:
Question 9:
The length of each side of an equilateral triangle of area , is
(a) 4 cm
(b)
(c)
(d) 3 cm
Answer:
Area of an equilateral triangle say A, having each side a cm is given by
We are asked to find the side of the triangle
Therefore, the side of the equilateral triangle says a, having area is given by
Therefore, the correct answer is (a).
Page No 180:
Question 10:
If an isosceles right triangle has area 8 cm2,then the length of its hypotenuse is
(a)
(b)
(c)
(d) 4 cm
Answer:
Given: Area of an isosceles right triangle is 8 cm2
Hence, the correct option is (a).
Page No 180:
Question 11:
The perimeter of an equilateral triangle is 60 m. The area is
(a)
(b)
(c)
(d)
Answer:
Given: The perimeter of an equilateral triangle is 60 m.
Let the length of the side of an equilateral triangle be x m.
Hence, the correct option is (d).
Page No 180:
Question 12:
The area of an isosceles triangle having base 2 cm and the length of one of the equal sides 4 cm, is
(a)
(b)
(c)
(d)
Answer:
Given:
The base of an isosceles triangle is 2 cm.
The length of one of the equal sides 4 cm.
Using Heron's formula:
If a, b and c are three sides of a triangle, then
area of the triangle = , where .
Here,
Hence, the correct option is (a).
Page No 180:
Question 13:
The length of each side of an equilateral triangle having an area of is
(a) 8 cm
(b) 36 cm
(c) 4 cm
(d) 6 cm
Answer:
Given: The area of an equilateral triangle is .
Let the length of the side of an equilateral triangle be x cm.
Hence, the correct option is (d).
Page No 180:
Question 14:
If the area of an equilateral triangle is , then its perimeter is
(a) 48 cm
(b) 24 cm
(c) 12 cm
(d) 36 cm
Answer:
Given: The area of an equilateral triangle is .
Let the length of the side of an equilateral triangle be x cm.
Hence, the correct option is (b).
Page No 180:
Question 15:
The sides of a triangle are 35 cm, 54 cm and 61 cm respectively. The length of its longest altitude is
(a)
(b)
(c)
(d) 28 cm
Answer:
Given:
The sides of a triangle are 35 cm, 54 cm and 61 cm respectively.
Let CD be the longest altitude of the triangle.
Using Heron's formula:
If a, b and c are three sides of a triangle, then
area of the triangle = , where .
Here,
We know,
Thus, the length of its longest altitude is .
Hence, the correct option is (c).
Page No 181:
Question 16:
The sides of a triangle are 56 cm, 60 cm and 52 cm. Area of the triangle is
(a) 1322 cm2
(b) 1311 cm2
(c) 1344 cm2
(d) 1392 cm2
Answer:
Given:
The sides of a triangle are 56 cm, 60 cm and 52 cm.
Using Heron's formula:
If a, b and c are three sides of a triangle, then
area of the triangle = , where .
Here,
Hence, the correct option is (c).
Page No 181:
Question 17:
The edges of a triangular board are 6 cm, 8 cm and 10 cm long. The cost of painting it at the rate of 9 paise per cm2 is
(a) â¹ 2
(b) â¹ 2.16
(c) â¹ 2.48
(d) â¹ 3
Answer:
Given:
The edges of a triangular board are 6 cm, 8 cm and 10 cm long.
The cost of painting per cm2 is 9 paise.
Using Heron's formula:
If a, b and c are three sides of a triangle, then
area of the triangle = , where .
Here,
The cost of painting per cm2 = 9 paise.
The cost of painting 24 cm2 = 24 × 9 paise
= 216 paise
= â¹ 2.16
Hence, the correct option is (b).
Page No 181:
Question 18:
The area of an equilateral triangle with side cm is
(a) 5.196 cm2
(b) 0.866 cm2
(c) 3.496 cm2
(d) 1.732 cm2
Answer:
Given: The side of an equilateral triangle is cm.
Hence, the correct option is (a).
Page No 181:
Question 19:
If the area of a regular hexagon is , then the length of its each side is
(a) 3 cm
(b)
(c) 6 cm
(d)
Answer:
Given: The area of a regular hexagon is .
We know, a hexagon is formed by joining 6 equilateral triangles together.
Therefore, Area of 6 equilateral triangles =
Hence, the correct option is (c).
Page No 181:
Question 20:
If the length of each edge of a regular tetrahedron is 'a', then its surface area is
(a)
(b)
(c)
(d)
Answer:
Given: The length of each edge of a regular tetrahedron is 'a' units.
We know, the surface area of tetrahedron = area of 4 equilateral triangles
Hence, the correct option is (a).
Page No 181:
Question 21:
If the area of an isosceles right triangle is 8 cm2, what is the perimeter of the triangle?
(a) 8 + cm2
(b) 8 + 4 cm2
(c) 4 + 8 cm2
(d) 12 cm2
Answer:
We are given the area of an isosceles right triangle and we have to find its perimeter.
Two sides of isosceles right triangle are equal and we assume the equal sides to be the base and height of the triangle. We are asked to find the perimeter of the triangle
Let us take the base and height of the triangle be x cm.
Area of a isosceles right triangle, say A having base x cm and height x cm is given by
A = 8 cm2; Base = Height = x cm
Using Pythagorean Theorem we have;
Let ABC be the given triangle
Perimeter of triangle ABC, say P is given by
AB = 4 cm; BC = 4 cm; AC =
Therefore, the answer is (b).
Page No 181:
Question 22:
The lengths of the sides of Δ ABC are consecutive integers. It Δ ABC has the same perimeter as an equilateral triangle with a side of length 9 cm, what is the length of the shortest side of ΔABC?
(a) 4
(b) 6
(c) 8
(d) 10
Answer:
We are given that triangle ABC has equal perimeter as to the perimeter of an equilateral triangle having side 9 cm. The sides of triangle ABC are consecutive integers. We are asked to find the smallest side of the triangle ABC
Perimeter of an equilateral triangle, say P having side 9 cm is given by
Let us assume the three sides of triangle ABC be x, x+1, x−1
Perimeter of triangle ABC, say P1 is given by
P1 = AB + BC + AC
AB = x; BC = x +1; AC = x−1. Since P1 = P. So
By using the value of x, we get the sides of triangle as 8 cm, 9 cm and 10 cm
Therefore, the answer is (c).
Page No 181:
Question 23:
In the given figure, the ratio AD to DC is 3 to 2. If the area of Δ ABC is 40 cm2, what is the area of Δ BDC?
(a) 16 cm2
(b) 24 cm2
(c) 30 cm2
(d) 36 cm2
Answer:
Area of triangle ABC is given 40 cm2.
Also
We are asked to find the area of the triangle BDC
Let us take BE perpendicular to base AC in triangle ABC.
We assume AC equal to y and BE equal to x in triangle ABC
Area of triangle ABC, say A is given by
We are given the ratio between AD to DC equal to 3:2
So,
In triangle BDC, we take BE as the height of the triangle
Area of triangle BDC, say A1 is given by
Therefore, the answer is (a).
Page No 181:
Question 24:
If the length of a median of an equilateral triangle is x cm, then its area is
(a) x2
(b)
(c)
(d)
Answer:
We are given the length of median of an equilateral triangle by which we can calculate its side. We are asked to find area of triangle in terms of x
Altitude of an equilateral triangle say L, having equal sides of a cm is given by, where, L = x cm
Area of an equilateral triangle, say A1 having each side a cm is given by
Since .So
Therefore, the answer is (c).
Page No 181:
Question 25:
If every side of a triangle is doubled, then increase in the area of the triangle is
(a)
(b) 200%
(c) 300%
(d) 400%
Answer:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
We take the sides of a new triangle as 2a, 2b, 2c that is twice the sides of previous one
Now, the area of a triangle having sides 2a, 2b, and 2c and as semi-perimeter is given by,
Where,
Now,
Therefore, increase in the area of the triangle
Percentage increase in area
Therefore, the answer is (c).
Page No 181:
Question 26:
A square and an equilateral triangle have equal perimeters. If the diagonal of the square is cm, then area of the triangle is
(a)
(b)
(c)
(d)
Answer:
It is given the perimeter of a square ABCD is equal to the perimeter of triangle PQR.
The measure of the diagonal of the square is given .We are asked to find the area of the triangle
In square ABCD, we assume that the adjacent sides of square be a.
Since, it is a square then
By using Pythagorean Theorem
Therefore, side of the square is 12 cm.
Perimeter of the square ABCD say P is given by
Side = 12 cm
Perimeter of the equilateral triangle PQR say P1 is given by
The side of equilateral triangle PQR is equal to 16 cm.
Area of an equilateral triangle say A, having each side a cm is given by
Area of the given equilateral triangle having each equal side equal to 4 cm is given by
a = 16 cm
Therefore, the answer is (d).
Page No 182:
Question 27:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The area â of an isosceles triangle with base a and each equal side b is given by .
Statement-2 (Reason): The area â of a triangle with semi-perimeter s and sides a, b and c is given by .
Answer:
Statement-2 (Reason): The area â of a triangle with semi-perimeter s and sides a, b and c is given by .
According to Heron's formula, the area of a triangle with sides a, b and c is given by where the semi-perimeter is .
Thus, Statement-2 is true.
Statement-1 (Assertion): The area â of an isosceles triangle with base a and each equal side b is given by .
Let an isosceles triangle with base a and each equal side b.
According to Statement-2,
And,
Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Hence, the correct answer is option (a).
Page No 182:
Question 28:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The altitude p of an equilateral triangle having each side a is given by .
Statement-2 (Reason): If p is the altitude of an equilateral triangle, then its area A is given by .
Answer:
Statement-2 (Reason): If p is the altitude of an equilateral triangle, then its area A is given by .
Let each side of the equilateral triangle be a and p be the altitude of an equilateral triangle.
According to Heron's formula, the area of a triangle with sides a, b and c is given by where the semi-perimeter is .
Here,
And,
Substituting (3) in (2), we get
Thus, Statement-2 is true.
Statement-1 (Assertion): The altitude p of an equilateral triangle having each side a is given by .
Let each side of the equilateral triangle be a and p be the altitude of an equilateral triangle.
According to Statement-2, if p is the altitude of an equilateral triangle, then its area A is given by
.....(1)
According to Heron's formula, the area of a triangle with sides a, b and c is given by where the semi-perimeter is .
Here,
And,
From (1) and (4), we get
Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Hence, the correct answer is option (a).
Page No 182:
Question 29:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The area of an equilateral triangle the length of whose altitude is 6 cm, is .
Statement-2 (Reason): The area of an equilateral triangle with altitude p is .
Answer:
Statement-2 (Reason): The area of an equilateral triangle with altitude p is .
Let each side of the equilateral triangle be a and p be the altitude of an equilateral triangle.
According to Heron's formula, the area of a triangle with sides a, b and c is given by where the semi-perimeter is .
Here,
And,
Substituting (3) in (2), we get
Thus, Statement-2 is true.
Statement-1 (Assertion): The area of an equilateral triangle the length of whose altitude is 6 cm, is .
Here, altitude of equilateral triangle (p) = 6 cm
According to Statement-2, if p is the altitude of an equilateral triangle, then its area A is given by
.....(1)
Now,
Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Hence, the correct answer is option (a).
Page No 182:
Question 30:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If the area of an equilateral triangle is , then its perimeter is 36 cm.
Statement-2 (Reason): If the perimeter of an equilateral triangle is 72 cm, then its altitude is cm.
Answer:
Statement-2 (Reason): If the perimeter of an equilateral triangle is 72 cm, then its altitude is cm.
Let each side of the equilateral triangle be a and p be the altitude of an equilateral triangle.
Given that, the perimeter of an equilateral triangle is 72 cm.
⇒ a + a + a = 72
⇒ 3a = 72
â⇒ a = 24
Since, the altitude p of an equilateral triangle having each side a is given by .
Thus, Statement-2 is false.
Statement-1 (Assertion): If the area of an equilateral triangle is , then its perimeter is 36 cm.
Let each side of the equilateral triangle be a.
Given that, the area of an equilateral triangle is .
Since, a is the length each side of the equilateral triangle then its area is given by .
Now, perimeter of the triangle = a + a + a
= 3a
= 3 × 12
= 36 cm
Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is false.
Hence, the correct answer is option (c).
Page No 182:
Question 31:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The area of an isosceles triangle having base 24 cm and each of the equal sides equal to 13 cm is 60 cm2.
Statement-2 (Reason): The area of an isosceles triangle with base a and each equal side b is .
Answer:
Statement-2 (Reason): The area of an isosceles triangle with base a and each equal side b is .
According to Heron's formula, the area of a triangle with sides a, b and c is given by where the semi-perimeter is .
Let an isosceles triangle with base a and each equal side b.
Here,
And,
Thus, Statement-2 is false.
Statement-1 (Assertion): The area of an isosceles triangle having base 24 cm and each of the equal sides equal to 13 cm is 60 cm2.
Given that, an isosceles triangle with base a = 24 cm and each equal side b = 13 cm.
Since, the area â of an isosceles triangle with base a and each equal side b is given by .
Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is false.
Hence, the correct answer is option (c).
Page No 182:
Question 32:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If the side of a rhombus is 10 cm and one diagonal is 16 cm, the area of the rhombus is 96 cm2.
Statement-2 (Reason): The base and the corresponding altitude of a parallelogram are 10 cm and 3.5 cm respectively. The area of the parallelogram is 30 cm2.
Answer:
Statement-2 (Reason): The base and the corresponding altitude of a parallelogram are 10 cm and 3.5 cm respectively. The area of the parallelogram is 30 cm2.
Given that, the base of a parallelogram (b) = 10 cm and the corresponding altitude of the parallelogram (h) = 3.5 cm.
âµ Area of a parallelogram = b × h
= 10 × 3.5
= 35 cm2
Thus, Statement-2 is false.
Statement-1 (Assertion): If the side of a rhombus is 10 cm and one diagonal is 16 cm, the area of the rhombus is 96 cm2.
Consider the ABCD be the rhombus such that AB = BC = CD = DA = 10 cm and AC = 16 cm.
According to Heron's formula, the area of a triangle with sides a, b and c is given by where the semi-perimeter is .
Here, a = AB = 10 cm, b = BC = 10 cm and c = AC = 16 cm.
And,
Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is false.
Hence, the correct answer is option (c).
View NCERT Solutions for all chapters of Class 9