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1. Find the area of ABC whose vertices are:
(i) 1 ,2 , 2,3 3, 4 A B and C
(ii)
5,7 , 4, 5 4,5 A B and C
(iii)
3,8 , 4,2 5, 1 A B and C
(iv) 
10, 6 , 2,5 1, 3 A B and C
Sol: 
(i) 1 ,2 , 2,3 3, 4 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
1, 2 , 2, 3 3, 4 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
1 3 4 2 4 2 3 2 3
2
1
1 3 4 2 6 3 1
2
1
7 12 3
2
1
22
2
11 .
x y y x y y x y y
sq units
(ii) 5,7 , 4, 5 4,5 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
5, 7 , 4, 5 4, 5 x y x y and x y
Page 2


1. Find the area of ABC whose vertices are:
(i) 1 ,2 , 2,3 3, 4 A B and C
(ii)
5,7 , 4, 5 4,5 A B and C
(iii)
3,8 , 4,2 5, 1 A B and C
(iv) 
10, 6 , 2,5 1, 3 A B and C
Sol: 
(i) 1 ,2 , 2,3 3, 4 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
1, 2 , 2, 3 3, 4 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
1 3 4 2 4 2 3 2 3
2
1
1 3 4 2 6 3 1
2
1
7 12 3
2
1
22
2
11 .
x y y x y y x y y
sq units
(ii) 5,7 , 4, 5 4,5 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
5, 7 , 4, 5 4, 5 x y x y and x y
 
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
5 5 5 4 5 7 4 7 5
2
1
5 10 4 2 4 12
2
1
50 8 48
2
1
106
2
53 .
x y y x y y x y y
sq units
(iii) 3,8 , 4,2 5, 1 A B and C are verticals of . ABC Then,
1 1 2 2 3 3
3, 8 , 4, 2 5, 1 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
x y y x y y x y y
1
3 2 1 4 1 8 5 8 2
2
1
3 2 1 4 9 5 6
2
1
9 36 30
2
1
75
2
37.5 . sq units
(iv)
10, 6 , 2,5 1, 3 A B and C
are the vertex of . ABC Then,
1 1 2 2 3 3
10, 6 , 2, 5 1, 3 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
10 5 3 2 3 6 1 6 5
2
x y y x y y x y y
1
10 2 2 9 1 11
2
1
20 18 11
2
Page 3


1. Find the area of ABC whose vertices are:
(i) 1 ,2 , 2,3 3, 4 A B and C
(ii)
5,7 , 4, 5 4,5 A B and C
(iii)
3,8 , 4,2 5, 1 A B and C
(iv) 
10, 6 , 2,5 1, 3 A B and C
Sol: 
(i) 1 ,2 , 2,3 3, 4 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
1, 2 , 2, 3 3, 4 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
1 3 4 2 4 2 3 2 3
2
1
1 3 4 2 6 3 1
2
1
7 12 3
2
1
22
2
11 .
x y y x y y x y y
sq units
(ii) 5,7 , 4, 5 4,5 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
5, 7 , 4, 5 4, 5 x y x y and x y
 
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
5 5 5 4 5 7 4 7 5
2
1
5 10 4 2 4 12
2
1
50 8 48
2
1
106
2
53 .
x y y x y y x y y
sq units
(iii) 3,8 , 4,2 5, 1 A B and C are verticals of . ABC Then,
1 1 2 2 3 3
3, 8 , 4, 2 5, 1 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
x y y x y y x y y
1
3 2 1 4 1 8 5 8 2
2
1
3 2 1 4 9 5 6
2
1
9 36 30
2
1
75
2
37.5 . sq units
(iv)
10, 6 , 2,5 1, 3 A B and C
are the vertex of . ABC Then,
1 1 2 2 3 3
10, 6 , 2, 5 1, 3 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
10 5 3 2 3 6 1 6 5
2
x y y x y y x y y
1
10 2 2 9 1 11
2
1
20 18 11
2
 
1
49
2
24.5 . sq units
2. Find the area of a quadrilateral ABCD whose vertices area A(3, -1), B(9, -5) C(14, 0) and
D(9, 19).
Sol:
By joining A and C, we get two triangles ABC and ACD.
Let
1 1 2 2 3 3 4 4
, 3, 1 , , 9, 5 , , 14,0 , 9,19 A x y A B x y B C x y C and D x y D
Then,
Area of 
1 2 3 2 3 1 3 1 2
1
2
ABC x y y x y y x y y
1
3 5 0 9 0 1 14 1 5
2
1
15 9 56 25 .
2
sq units
Area of 
1 3 4 3 4 1 4 1 3
1
2
ACD x y y x y y x y y
1
3 0 19 14 19 1 9 1 0
2
1
57 280 9 107 .
2
sq units
So, the area of the quadrilateral is 25 107 132 . . sq units
3. Find the area of quadrilateral PQRS whose vertices are P(-5, -3), Q(-4,-6),R(2, -3) and
S(1,2).
Sol:
By joining P and R, we get two triangles PQR and PRS.
Let 
1 1 2 2 3 3
, 5, 3 , , 4, 6 , , 2, 3 P x y P Q x y Q R x y R and. Then 
4 4
, 1,2 S x y S
Area of 
1 2 3 2 3 1 3 1 2
1
2
PQR x y y x y y x y y
1
5 6 3 4 3 3 2 3 6
2
1 21
15 0 6 .
2 2
sq units
Page 4


1. Find the area of ABC whose vertices are:
(i) 1 ,2 , 2,3 3, 4 A B and C
(ii)
5,7 , 4, 5 4,5 A B and C
(iii)
3,8 , 4,2 5, 1 A B and C
(iv) 
10, 6 , 2,5 1, 3 A B and C
Sol: 
(i) 1 ,2 , 2,3 3, 4 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
1, 2 , 2, 3 3, 4 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
1 3 4 2 4 2 3 2 3
2
1
1 3 4 2 6 3 1
2
1
7 12 3
2
1
22
2
11 .
x y y x y y x y y
sq units
(ii) 5,7 , 4, 5 4,5 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
5, 7 , 4, 5 4, 5 x y x y and x y
 
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
5 5 5 4 5 7 4 7 5
2
1
5 10 4 2 4 12
2
1
50 8 48
2
1
106
2
53 .
x y y x y y x y y
sq units
(iii) 3,8 , 4,2 5, 1 A B and C are verticals of . ABC Then,
1 1 2 2 3 3
3, 8 , 4, 2 5, 1 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
x y y x y y x y y
1
3 2 1 4 1 8 5 8 2
2
1
3 2 1 4 9 5 6
2
1
9 36 30
2
1
75
2
37.5 . sq units
(iv)
10, 6 , 2,5 1, 3 A B and C
are the vertex of . ABC Then,
1 1 2 2 3 3
10, 6 , 2, 5 1, 3 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
10 5 3 2 3 6 1 6 5
2
x y y x y y x y y
1
10 2 2 9 1 11
2
1
20 18 11
2
 
1
49
2
24.5 . sq units
2. Find the area of a quadrilateral ABCD whose vertices area A(3, -1), B(9, -5) C(14, 0) and
D(9, 19).
Sol:
By joining A and C, we get two triangles ABC and ACD.
Let
1 1 2 2 3 3 4 4
, 3, 1 , , 9, 5 , , 14,0 , 9,19 A x y A B x y B C x y C and D x y D
Then,
Area of 
1 2 3 2 3 1 3 1 2
1
2
ABC x y y x y y x y y
1
3 5 0 9 0 1 14 1 5
2
1
15 9 56 25 .
2
sq units
Area of 
1 3 4 3 4 1 4 1 3
1
2
ACD x y y x y y x y y
1
3 0 19 14 19 1 9 1 0
2
1
57 280 9 107 .
2
sq units
So, the area of the quadrilateral is 25 107 132 . . sq units
3. Find the area of quadrilateral PQRS whose vertices are P(-5, -3), Q(-4,-6),R(2, -3) and
S(1,2).
Sol:
By joining P and R, we get two triangles PQR and PRS.
Let 
1 1 2 2 3 3
, 5, 3 , , 4, 6 , , 2, 3 P x y P Q x y Q R x y R and. Then 
4 4
, 1,2 S x y S
Area of 
1 2 3 2 3 1 3 1 2
1
2
PQR x y y x y y x y y
1
5 6 3 4 3 3 2 3 6
2
1 21
15 0 6 .
2 2
sq units
 
Area of 
1 3 4 3 4 1 4 1 3
1
2
PRS x y y x y y x y y
1
5 3 2 2 2 3 1 3 3
2
1 35
25 10 0 .
2 2
sq units
So, the area of the quadrilateral PQRS is
21 35
28 . .
2 2
sq units sq units
4. Find the area of quadrilateral ABCD whose vertices are A(-3, -1), B(-2,-4) C(4,-1) and
D(3,4)
Sol:
By joining A and C, we get two triangles ABC and ACD.
Let 
1 1 2 2 3 3
, 3, 1 , , 2, 4 , , 4, 1 A x y A B x y B C x y C and. Then 
4 4
, 3,4 D x y D
Area of 
1 2 3 2 3 1 3 1 2
1
2
ABC x y y x y y x y y
1
3 4 1 2 1 1 4 1 4
2
1 21
9 0 12 .
2 2
sq units
Area of 
1 3 4 3 4 1 4 1 3
1
2
ACD x y y x y y x y y
1
3 1 4 4 4 1 3 1 1
2
1 35
15 20 0 .
2 2
sq units
So, the area of the quadrilateral ABCD is
21 35
28 . .
2 2
sq units sq units
5. Find the area of quadrilateral ABCD whose vertices are A(-5, 7), B(-4, -5) C(-1,-6) and
D(4,5)
Sol:
By joining A and C, we get two triangles ABC and ACD.
Let 
1 1 2 2 3 3
, 5,7 , , 4, 5 , , 1 , 6 A x y A B x y B C x y C and. 
4 4
, 4,5 D x y D
Then 
Page 5


1. Find the area of ABC whose vertices are:
(i) 1 ,2 , 2,3 3, 4 A B and C
(ii)
5,7 , 4, 5 4,5 A B and C
(iii)
3,8 , 4,2 5, 1 A B and C
(iv) 
10, 6 , 2,5 1, 3 A B and C
Sol: 
(i) 1 ,2 , 2,3 3, 4 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
1, 2 , 2, 3 3, 4 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
1 3 4 2 4 2 3 2 3
2
1
1 3 4 2 6 3 1
2
1
7 12 3
2
1
22
2
11 .
x y y x y y x y y
sq units
(ii) 5,7 , 4, 5 4,5 A B and C are the vertices of . ABC Then,
1 1 2 2 3 3
5, 7 , 4, 5 4, 5 x y x y and x y
 
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
5 5 5 4 5 7 4 7 5
2
1
5 10 4 2 4 12
2
1
50 8 48
2
1
106
2
53 .
x y y x y y x y y
sq units
(iii) 3,8 , 4,2 5, 1 A B and C are verticals of . ABC Then,
1 1 2 2 3 3
3, 8 , 4, 2 5, 1 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
x y y x y y x y y
1
3 2 1 4 1 8 5 8 2
2
1
3 2 1 4 9 5 6
2
1
9 36 30
2
1
75
2
37.5 . sq units
(iv)
10, 6 , 2,5 1, 3 A B and C
are the vertex of . ABC Then,
1 1 2 2 3 3
10, 6 , 2, 5 1, 3 x y x y and x y
Area of triangle ABC
1 2 3 2 3 1 3 1 2
1
2
1
10 5 3 2 3 6 1 6 5
2
x y y x y y x y y
1
10 2 2 9 1 11
2
1
20 18 11
2
 
1
49
2
24.5 . sq units
2. Find the area of a quadrilateral ABCD whose vertices area A(3, -1), B(9, -5) C(14, 0) and
D(9, 19).
Sol:
By joining A and C, we get two triangles ABC and ACD.
Let
1 1 2 2 3 3 4 4
, 3, 1 , , 9, 5 , , 14,0 , 9,19 A x y A B x y B C x y C and D x y D
Then,
Area of 
1 2 3 2 3 1 3 1 2
1
2
ABC x y y x y y x y y
1
3 5 0 9 0 1 14 1 5
2
1
15 9 56 25 .
2
sq units
Area of 
1 3 4 3 4 1 4 1 3
1
2
ACD x y y x y y x y y
1
3 0 19 14 19 1 9 1 0
2
1
57 280 9 107 .
2
sq units
So, the area of the quadrilateral is 25 107 132 . . sq units
3. Find the area of quadrilateral PQRS whose vertices are P(-5, -3), Q(-4,-6),R(2, -3) and
S(1,2).
Sol:
By joining P and R, we get two triangles PQR and PRS.
Let 
1 1 2 2 3 3
, 5, 3 , , 4, 6 , , 2, 3 P x y P Q x y Q R x y R and. Then 
4 4
, 1,2 S x y S
Area of 
1 2 3 2 3 1 3 1 2
1
2
PQR x y y x y y x y y
1
5 6 3 4 3 3 2 3 6
2
1 21
15 0 6 .
2 2
sq units
 
Area of 
1 3 4 3 4 1 4 1 3
1
2
PRS x y y x y y x y y
1
5 3 2 2 2 3 1 3 3
2
1 35
25 10 0 .
2 2
sq units
So, the area of the quadrilateral PQRS is
21 35
28 . .
2 2
sq units sq units
4. Find the area of quadrilateral ABCD whose vertices are A(-3, -1), B(-2,-4) C(4,-1) and
D(3,4)
Sol:
By joining A and C, we get two triangles ABC and ACD.
Let 
1 1 2 2 3 3
, 3, 1 , , 2, 4 , , 4, 1 A x y A B x y B C x y C and. Then 
4 4
, 3,4 D x y D
Area of 
1 2 3 2 3 1 3 1 2
1
2
ABC x y y x y y x y y
1
3 4 1 2 1 1 4 1 4
2
1 21
9 0 12 .
2 2
sq units
Area of 
1 3 4 3 4 1 4 1 3
1
2
ACD x y y x y y x y y
1
3 1 4 4 4 1 3 1 1
2
1 35
15 20 0 .
2 2
sq units
So, the area of the quadrilateral ABCD is
21 35
28 . .
2 2
sq units sq units
5. Find the area of quadrilateral ABCD whose vertices are A(-5, 7), B(-4, -5) C(-1,-6) and
D(4,5)
Sol:
By joining A and C, we get two triangles ABC and ACD.
Let 
1 1 2 2 3 3
, 5,7 , , 4, 5 , , 1 , 6 A x y A B x y B C x y C and. 
4 4
, 4,5 D x y D
Then 
 
Area of 
1 2 3 2 3 1 3 1 2
1
2
ABC x y y x y y x y y
1
5 5 6 4 6 7 1 7 5
2
1 35
5 52 12 .
2 2
sq units
Area of 
1 3 4 3 4 1 4 1 3
1
2
ACD x y y x y y x y y
1
5 6 5 1 5 7 4 7 6
2
1 109
55 2 52 .
2 2
sq units
So, the area of the quadrilateral ABCD is
35 109
72 .
2 2
sq units
6. Find the area of the triangle formed by joining the midpoints of the sides of the triangle
whose vertices are A(2,1) B(4,3) and C(2,5)
Sol:
The verticals of the triangle are 2,1 , 4,3 2,5 . A B and C
Coordinates of midpoint of 
1 1
2 4 1 3
, , 3, 2
2 2
AB P x y
Coordinates of midpoint of 
2 2
4 2 3 5
, , 3,4
2 2
BC Q x y
Coordinates of midpoint of 
3 3
2 2 1 5
, , 2,3
2 2
AC R x y
Now,
Area of 
2 2 3 2 3 1 3 1 2
1
2
PQR x y y x y y x y y
1
3 4 3 3 3 2 2 2 4
2
1
3 3 4 1 .
2
sq unit
Hence, the area of the quadrilateral triangle is 1 sq. unit.
7. A(7, -3), B(5,3) and C(3,-1) are the vertices of a ABC and AD is its median. Prove that
the median AD divides ABC into two triangles of equal areas.
Sol:
The vertices of the triangle are 7, 3 , 5,3 , 3, 1 . A B C

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