NCERT Exercise Solutions provided here ensure that students get all the vital information on calculating the Areas of a triangle. Areas of a triangle are very important from the point of CBSE Class 10 final exams. Hence, students are advised to get familiar with the formulas and methods of calculating areas of triangles using the NCERT Class 10 Maths Solutions.
A great way of learning Maths is by solving problems. To help students learn the methods of calculating the areas of the triangle, we have provided detailed NCERT solutions to questions provided in Exercise 7.3 of NCERT Class 10 textbook.
Topics Covered in Exercise 7.3
- Section formula
- Solved examples
How Is It Helpful
- Offers in-depth knowledge of solving problems in the areas of triangles.
- NCERT Class 10 Solutions guides students in clearing all the fundamental concepts.
- Offers different types of questions to practise.
NCERT Solutions for Class 10 Maths Chapter 7- Coordinate Geometry Exercise 7.3
Access Other Exercise Solutions of Class 10 Maths Chapter 7 – Coordinate Geometry
By clicking on the links below, students will be able to access the other exercise of NCERT Solutions for Class 10 Maths Chapter 7.
Exercise 7.1 – 10 Questions (8 practical-based questions, 2 reasoning questions)
Exercise 7.2 – 10 Questions (8 long answer questions, 2 short answer questions)
Exercise 7.4 – 8 Questions (6 long answer questions, 1 practical-based question, 1 reasoning question)
Access Answers to NCERT Class 10 Maths Chapter 7 – Coordinate Geometry Exercise 7.3
Exercise 7.3 Page No: 170
1. Find the area of the triangle whose vertices are
(i) (2, 3), (-1, 0), (2, -4)
(ii) (-5, -1), (3, -5), (5, 2)
Solution:
Area of a triangle formula = 1/2 × [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)]
(i) Here,
x1 = 2, x2 = -1, x3 = 2, y1 = 3, y2 = 0 and y3 = -4
Substitute all the values in the above formula, and we get
Area of triangle = 1/2 [2 {0- (-4)} + (-1) {(-4) – (3)} + 2 (3 – 0)]
= 1/2 {8 + 7 + 6}
= 21/2
So, the area of the triangle is 21/2 square units.
(ii) Here,
x1 = -5, x2 = 3, x3 = 5, y1 = -1, y2 = -5 and y3 = 2
Area of the triangle = 1/2 [-5 { (-5)- (2)} + 3(2-(-1)) + 5{-1 – (-5)}]
= 1/2{35 + 9 + 20} = 32
Therefore, the area of the triangle is 32 square units.
2. In each of the following, find the value of ‘k’, for which the points are collinear.
(i) (7, -2), (5, 1), (3, -k)
(ii) (8, 1), (k, -4), (2, -5)
Solution:
(i) For collinear points, the area of the triangle formed by them is always zero.
Let points (7, -2), (5, 1) and (3, k) be vertices of a triangle.
Area of triangle = 1/2 [7 { 1- k} + 5(k-(-2)) + 3{(-2) – 1}] = 0
7 – 7k + 5k +10 -9 = 0
-2k + 8 = 0
k = 4
(ii) For collinear points, the area of the triangle formed by them is zero.
Therefore, for points (8, 1), (k, – 4), and (2, – 5), area = 0
1/2 [8 { -4- (-5)} + k{(-5)-(1)} + 2{1 -(-4)}] = 0
8 – 6k + 10 = 0
6k = 18
k = 3
3. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.
Solution:
Let the vertices of the triangle be A (0, -1), B (2, 1), and C (0, 3).
Let D, E, and F be the mid-points of the sides of this triangle.
Coordinates of D, E, and F are given by
D = (0+2/2, -1+1/2 ) = (1, 0)
E = ( 0+0/2, -1+3/2 ) = (0, 1)
F = ( 0+2/2, 3+1/2 ) = (1, 2)
Area of a triangle = 1/2 × [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)]
Area of ΔDEF = 1/2 {1(2-1) + 1(1-0) + 0(0-2)} = 1/2 (1+1) = 1
The area of ΔDEF is 1 square unit
Area of ΔABC = 1/2 [0(1-3) + 2{3-(-1)} + 0(-1-1)] = 1/2 {8} = 4
The area of ΔABC is 4 square units
Therefore, the required ratio is 1:4.
4. Find the area of the quadrilateral whose vertices, taken in order, are
(-4, -2), (-3, -5), (3, -2) and (2, 3).
Solution:
Let the vertices of the quadrilateral be A (- 4, – 2), B ( – 3, – 5), C (3, – 2), and D (2, 3).
Join AC and divide the quadrilateral into two triangles.
We have two triangles, ΔABC and ΔACD.
Area of a triangle = 1/2 × [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)]
Area of ΔABC = 1/2 [(-4) {(-5) – (-2)} + (-3) {(-2) – (-2)} + 3 {(-2) – (-5)}]
= 1/2 (12 + 0 + 9)
= 21/2 square units
Area of ΔACD = 1/2 [(-4) {(-2) – (3)} + 3{(3) – (-2)} + 2 {(-2) – (-2)}]
= 1/2 (20 + 15 + 0)
= 35/2 square units
Area of quadrilateral ABCD = Area of ΔABC + Area of ΔACD
= (21/2 + 35/2) square units = 28 square units
5. You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC, whose vertices are A (4, – 6), B (3, – 2) and C (5, 2).
Solution:
Let the vertices of the triangle be A (4, -6), B (3, -2), and C (5, 2).
Let D be the mid-point of side BC of ΔABC. Therefore, AD is the median in ΔABC.
Coordinates of point D = Midpoint of BC = ((3+5)/2, (-2+2)/2) = (4, 0)
Formula to find the area of a triangle = 1/2 × [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)]
Now, the Area of ΔABD = 1/2 [(4) {(-2) – (0)} + 3{(0) – (-6)} + (4) {(-6) – (-2)}]
= 1/2 (-8 + 18 – 16)
= -3 square units
However, the area cannot be negative. Therefore, the area of ΔABD is 3 square units.
Area of ΔACD = 1/2 [(4) {0 – (2)} + 4{(2) – (-6)} + (5) {(-6) – (0)}]
= 1/2 (-8 + 32 – 30) = -3 square units
However, the area cannot be negative. Therefore, the area of ΔACD is 3 square units.
The area of both sides is the same. Thus, median AD has divided ΔABC into two triangles of equal areas.
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