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COORDINATE GEOMETRY

  • Sum based on Finding Area of a triangle

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Q. You have studied in class IX, (Chapter 9, Example 3),

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).

Sol.

D is the mid-point of BC.

∴

By Midpoint formula,

∴ Coordinates of D

=

3

+

5

2

,

–2

+

2

2

D

=

, 0

8

2

D

=

(4,0)

A (4, –6)

​

B(3, –2)

C (5, 2)

​

D

Which formula is used to find co-ordinates of midpoint of a segment

x1 = 3,

y1 = –2

x2 = 5,

y2 = 2

Let the co-ordinates of B be (x1, y1)

Let the co-ordinates of C be (x2, y2)

Mid-Point Formula

,

+

x1

x2

2

+

y1

y2

2

(4,0)

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Q. You have studied in class IX, (Chapter 9, Example 3),

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).

Sol.

Area of triangle cannot be negative.

ar (ΔABD)

=

[

4

1

2

(–2

–

0)

+

3

(0

+

6)

+

4

(–6

+

2)]

Given :

A

(4, –6) ,

B

(3, –2),

D

(4, 0)

A (4, –6)

​

B (3, –2)

C (5, 2)

​

D

(4,0)

For ΔABD, the co-ordinates of A,B,D

x1 = 4,

y1 = –6

x2 = 3,

y2 = –2

x3 = 4,

y3 = 0

1

2

=

(–8

+

18

–

16)

1

2

=

(– 6)

=

–3

×

∴

ar (ΔABD)

=

3 sq. units

Let the co-ordinates of D be (x3, y3)

Let the co-ordinates of B be (x2, y2)

Let the co-ordinates of A be (x1, y1)

Consider ΔABD

Which formula is used to find area of triangle?

[x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]

1

2

–3

4 of 4

(– 6)

×

Q. You have studied in class IX, (Chapter 9, Example 3),

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).

Sol.

ar (ΔADC)

=

[

4

1

2

(0

–

2)

+

4

(2

+

6)

+

5

(–6

–

0)]

Given :

A

(4, –6),

D

(4, 0),

C

(5, 2)

A (4, –6)

​

B(3, –2)

C (5, 2)

​

D

(4,0)

For ΔADC, the co-ordinates of A,D,C

x1 = 4,

y1 = –6

x2 = 4,

y2 = 0

x3 = 5,

y3 = 2

Let the co-ordinates of C be (x3, y3)

Let the co-ordinates of D be (x2, y2)

Let the co-ordinates of A be (x1, y1)

1

2

=

(–8

+

32

1

2

=

=

– 3 sq. units

–

30)

∴

ar (ΔADC)

=

3 sq. units

∴

ar (ΔABD)

=

ar (ΔADC)

Consider ΔADC

Which formula is used to find area of triangle?

[x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]

1

2

–3

Area of triangle cannot be negative.