## NCERT Solutions for Class 10 Maths Exercise 7.3 Chapter 7 Coordinate Geometry â€“ FREE PDF Download

NCERT Class 10 Maths Ch 7 is one of the most important ones in the NCERT syllabus. Duly following NCERT Solutions for Class 10 Maths Chapter

7 ensures you that there will be no hindrance when you opt for more advanced branches of Maths. This is where CoolGyan comes in. Our free Class 10 Coordinate Geometry solutions will help you understand the chapter thoroughly.

# NCERT Solutions for Class 10 Maths Chapter 7 â€“ Coordinate Geometry

1. Find the area of the triangle whose vertices are:

(i) (2, 3), (â€“1, 0), (2, â€“4)

(ii) (â€“5, â€“1), (3, â€“5), (5, 2)

Area of Triangle =Â

=Â [2 {0 âˆ’ (âˆ’4)} â€“ 1 (âˆ’4 âˆ’ 3) + 2 (3 âˆ’ 0)]

=Â [2 (0 + 4) â€“ 1 (âˆ’7) + 2 (3)]

=Â (8 + 7 + 6) =Â sq. units

(ii)Â (â€“5, â€“1), (3, â€“5), (5, 2)

Area of Triangle =Â

= Â [âˆ’5 (âˆ’5 âˆ’ 2) + 3 {2 âˆ’ (âˆ’1)} + 5 {âˆ’1 âˆ’ (âˆ’5)}]

=Â Â [âˆ’5 (âˆ’7) + 3 (3) + 5 (4)]

=Â Â (35 + 9 + 20)

=Â Â (64) = 32 sq. units

(i) (2, 3), (â€“1, 0), (2, â€“4)

(ii) (â€“5, â€“1), (3, â€“5), (5, 2)

**Ans.**Â (i)Â (2, 3), (â€“1, 0), (2, â€“4)

Area of Triangle =Â

=Â [2 {0 âˆ’ (âˆ’4)} â€“ 1 (âˆ’4 âˆ’ 3) + 2 (3 âˆ’ 0)]

=Â [2 (0 + 4) â€“ 1 (âˆ’7) + 2 (3)]

=Â (8 + 7 + 6) =Â sq. units

(ii)Â (â€“5, â€“1), (3, â€“5), (5, 2)

Area of Triangle =Â

= Â [âˆ’5 (âˆ’5 âˆ’ 2) + 3 {2 âˆ’ (âˆ’1)} + 5 {âˆ’1 âˆ’ (âˆ’5)}]

=Â Â [âˆ’5 (âˆ’7) + 3 (3) + 5 (4)]

=Â Â (35 + 9 + 20)

=Â Â (64) = 32 sq. units

2. In each of the following find the value of â€˜kâ€™, for which the points are collinear.

(i) (7, â€“2), (5, 1), (3, k)

(ii) (8, 1), (k, â€“4), (2, â€“5)

Since, the given points are collinear, it means the area of triangle formed by them is equal to zero.

Area of Triangle =Â

â‡’Â Â [7 (1 âˆ’ k) + 5 {k âˆ’ (âˆ’2)} + 3 (âˆ’2 âˆ’ 1)] = 0

=Â Â (7 âˆ’ 7k + 5k + 10 âˆ’ 9) = 0

â‡’Â Â (7 âˆ’ 7k + 5k + 1) = 0

â‡’Â Â (8 âˆ’ 2k) = 0

â‡’ 8 âˆ’ 2k = 0

â‡’ 2k = 8

â‡’ k = 4

(ii)Â (8, 1), (k, â€“4), (2, â€“5)

Since, the given points are collinear, it means the area of triangle formed by them is equal to zero.

Area of Triangle =Â

â‡’Â Â [8 {âˆ’4 âˆ’ (âˆ’5)} + k (âˆ’5 âˆ’ 1) + 2 {1 âˆ’ (âˆ’4)}] = 0

â‡’â‡’Â Â (8 âˆ’ 6k + 10) = 0

â‡’Â (18 âˆ’ 6k) = 0

â‡’18 âˆ’ 6k = 0

â‡’ 18 = 6k

â‡’ k = 3

(i) (7, â€“2), (5, 1), (3, k)

(ii) (8, 1), (k, â€“4), (2, â€“5)

**Ans.**Â (i)Â (7, â€“2), (5, 1), (3, k)

Since, the given points are collinear, it means the area of triangle formed by them is equal to zero.

Area of Triangle =Â

â‡’Â Â [7 (1 âˆ’ k) + 5 {k âˆ’ (âˆ’2)} + 3 (âˆ’2 âˆ’ 1)] = 0

=Â Â (7 âˆ’ 7k + 5k + 10 âˆ’ 9) = 0

â‡’Â Â (7 âˆ’ 7k + 5k + 1) = 0

â‡’Â Â (8 âˆ’ 2k) = 0

â‡’ 8 âˆ’ 2k = 0

â‡’ 2k = 8

â‡’ k = 4

(ii)Â (8, 1), (k, â€“4), (2, â€“5)

Since, the given points are collinear, it means the area of triangle formed by them is equal to zero.

Area of Triangle =Â

â‡’Â Â [8 {âˆ’4 âˆ’ (âˆ’5)} + k (âˆ’5 âˆ’ 1) + 2 {1 âˆ’ (âˆ’4)}] = 0

â‡’â‡’Â Â (8 âˆ’ 6k + 10) = 0

â‡’Â (18 âˆ’ 6k) = 0

â‡’18 âˆ’ 6k = 0

â‡’ 18 = 6k

â‡’ k = 3

3. Find the area of the triangle formed by joining the midâ€“points of the sides of the triangle whose vertices are (0, â€“1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

C = (0, 3) =Â

Area of â–³ABC =Â

â‡’ Area of â–³ABC

=Â Â [0 (1 âˆ’ 3) + 2 {3 âˆ’ (âˆ’1)} + 0 (âˆ’1 âˆ’ 1)] =Â

= 4 sq. units

P, Q and R are the midâ€“points of sides AB, AC and BC respectively.

Applying Section Formula to find the vertices of P, Q and R, we get

Applying same formula, Area of â–³PQR =Â Â [1 (1 âˆ’ 2) + 0 (2 âˆ’ 0) + 1 (0 âˆ’ 1)] =Â Â

= 1 sq. units (numerically)

Now,Â

**Ans.**Â Let A = (0, â€“1) = , B = (2, 1) =Â and

C = (0, 3) =Â

Area of â–³ABC =Â

â‡’ Area of â–³ABC

=Â Â [0 (1 âˆ’ 3) + 2 {3 âˆ’ (âˆ’1)} + 0 (âˆ’1 âˆ’ 1)] =Â

= 4 sq. units

P, Q and R are the midâ€“points of sides AB, AC and BC respectively.

Applying Section Formula to find the vertices of P, Q and R, we get

Applying same formula, Area of â–³PQR =Â Â [1 (1 âˆ’ 2) + 0 (2 âˆ’ 0) + 1 (0 âˆ’ 1)] =Â Â

= 1 sq. units (numerically)

Now,Â

4. Find the area of the quadrilateral whose vertices taken in order are (â€“4, â€“2), (â€“3, â€“5), (3, â€“2) and (2, 3).

= Area of Triangle ABD +

Area of Triangle BCD â€¦ (1)

Using formula to find area of triangle:

Area ofÂ ABD

=Â

= Â [âˆ’4 (âˆ’5 âˆ’ 3) â€“ 3 {3 âˆ’ (âˆ’2)} + 2 {âˆ’2 âˆ’ (âˆ’5)}]

=Â Â (32 â€“ 15 + 6)

=Â Â (23) = 11.5 sq units â€¦ (2)

Again using formula to find area of triangle:

Area of â–³BCD =Â

=Â Â [âˆ’3 (âˆ’2 âˆ’ 3) + 3 {3 âˆ’ (âˆ’5)} + 2 {âˆ’5 âˆ’ (âˆ’2)}]

=Â Â (15 + 24 âˆ’ 6)

=Â Â (33) = 16.5 sq units â€¦ (3)

Putting (2) and (3) in (1), we get

Area of Quadrilateral ABCD = 11.5 + 16.5 = 28 sq units

**Ans.**Â Area of Quadrilateral ABCD

= Area of Triangle ABD +

Area of Triangle BCD â€¦ (1)

Using formula to find area of triangle:

Area ofÂ ABD

=Â

= Â [âˆ’4 (âˆ’5 âˆ’ 3) â€“ 3 {3 âˆ’ (âˆ’2)} + 2 {âˆ’2 âˆ’ (âˆ’5)}]

=Â Â (32 â€“ 15 + 6)

=Â Â (23) = 11.5 sq units â€¦ (2)

Again using formula to find area of triangle:

Area of â–³BCD =Â

=Â Â [âˆ’3 (âˆ’2 âˆ’ 3) + 3 {3 âˆ’ (âˆ’5)} + 2 {âˆ’5 âˆ’ (âˆ’2)}]

=Â Â (15 + 24 âˆ’ 6)

=Â Â (33) = 16.5 sq units â€¦ (3)

Putting (2) and (3) in (1), we get

Area of Quadrilateral ABCD = 11.5 + 16.5 = 28 sq units

5. We know that median of a triangle divides it into two triangles of equal areas. Verify this result for â–³ABC whose vertices are A (4, â€“6), B (3, â€“2) and C (5, 2).

We need to show that ar(â–³ABD) = ar(â–³ACD).

Let coordinates of point D are (x, y)

Using section formula to find coordinates of D, we get

Therefore, coordinates of point D are (4, 0)

Using formula to find area of triangle:

Area of â–³ABD =Â

=Â Â [4 (âˆ’2 âˆ’ 0) + 3 {0 âˆ’ (âˆ’6)} + 4 {âˆ’6 âˆ’ (âˆ’2)}]

= Â (âˆ’8 + 18 âˆ’16)

= Â (âˆ’6) = âˆ’3 sq units

Area cannot be in negative.

Therefore, we just consider its numerical value.

Therefore, area of â–³ABD = 3 sq units â€¦ (1)

Again using formula to find area of triangle:

Area of â–³ACD =Â

=Â Â [4 (2 âˆ’ 0) + 5 {0 âˆ’ (âˆ’6)} + 4 {âˆ’6 âˆ’2 )}]

=Â Â (8 + 30 âˆ’ 32) = Â½ (6) = 3 sq units â€¦ (2)

From (1) and (2), we get ar(â–³ABD) = ar(â–³ACD)

Hence Proved.

**Ans.**Â We have â–³ABC whose vertices are given.

We need to show that ar(â–³ABD) = ar(â–³ACD).

Let coordinates of point D are (x, y)

Using section formula to find coordinates of D, we get

Therefore, coordinates of point D are (4, 0)

Using formula to find area of triangle:

Area of â–³ABD =Â

=Â Â [4 (âˆ’2 âˆ’ 0) + 3 {0 âˆ’ (âˆ’6)} + 4 {âˆ’6 âˆ’ (âˆ’2)}]

= Â (âˆ’8 + 18 âˆ’16)

= Â (âˆ’6) = âˆ’3 sq units

Area cannot be in negative.

Therefore, we just consider its numerical value.

Therefore, area of â–³ABD = 3 sq units â€¦ (1)

Again using formula to find area of triangle:

Area of â–³ACD =Â

=Â Â [4 (2 âˆ’ 0) + 5 {0 âˆ’ (âˆ’6)} + 4 {âˆ’6 âˆ’2 )}]

=Â Â (8 + 30 âˆ’ 32) = Â½ (6) = 3 sq units â€¦ (2)

From (1) and (2), we get ar(â–³ABD) = ar(â–³ACD)

Hence Proved.