## RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.4

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.4

**Other Exercises**

- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.1
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.2
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.3
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.4
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.5
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry VSAQS
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry MCQS

**Question 1.**

Find the centroid of the triangle whose vertices are :

(i) (1, 4), (-1, -1), (3, -2)

(ii) (-2, 3), (2, -1), (4, 0)

**Solution:**

**Question 2.**

Two vertices of a triangle are (1, 2), (3, 5) and its centroid is at the origin. Find the Co-ordinates of the third vertex.

**Solution:**

Centroid of a triangle is O(0, 0) ….(i)

Co-ordinates of two vertices of a ∆ABC are A (1, 2) and B (3, 5)

Let the third vertex be (x, y)

**Question 3.**

Find the third vertex of a triangle, if two of its vertices are at (-3, 1) and (0, -2) and the centroid is at the origin.

**Solution:**

Let two vertices of a ∆ABC be A (-3, 1) and B (0, -2) and third vertex C be (x, y)

Centroid of the ∆ABC is O (0, 0)

**Question 4.**

A (3, 2) and B (-2, 1) are two vertices of a triangle ABC whose centroid G has the coordinates ( , ) . Find the coordinates of the third vertex C of the triangle. **[CBSE 2004]**

**Solution:**

A (3, 2) and B (-2, 1) are the two vertices of ∆ABC whose centroid is G ( , )

Let third vertex C be (x, y)

**Question 5.**

If (-2, 3), (4, -3) and (4, 5) are the mid-points of the sides of a triangle, find the co-ordinates of its centroid.

**Solution:**

In ∆ABC, D, E and F are the mid-points of the sides BC, CA and AB respectively.

The co-ordinates of D are (-2, 3), of E are (4,-3) and of F are (4, 5)

Let the co-ordinates of A, B and C be (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) respectively

**Question 6.**

Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.

**Solution:**

In ∆ABC,

D and E are the mid points of the sides AB and AC respectively

**Question 7.**

Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another.

**Solution:**

Let A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3}) and D (x_{4}, y_{4}) be the vertices of quadrilateral ABCD

E and F are the mid points of side BC and AD respectively and EF is joined G and H are the mid points of diagonal AC and BD.

GH are joined

**Question 8.**

If G be the centroid of a triangle ABC and P be any other point in the plane, prove that PA² + PB² + PC² = GA² + GB² + GC² + 3GP².

**Solution:**

In AABC, G is the centroid of it Let P (h, x) is any point in the plane

Let co-ordinates of A are (x_{1}, y_{1}) of B are (x_{2}, y_{2}) and of C are (x_{3}, y_{3})

Hence proved.

**Question 9.**

If G be the centroid of a triangle ABC, prove that AB² + BC² + CA² = 3 (GA² + GB² + GC²)

**Solution:**

Let the co-ordinates of the vertices of ∆ABC be A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3}) and let G be the centroid of the triangle

Hence proved.

**Question 10.**

In the figure, a right triangle BOA is given. C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A and B.

**Solution:**

In right ∆OAB, co-ordinates of O are (0, 0) of A are (2a, 0) and of B are (0, 2b)

C is the mid-point of AB

Co-ordinates of C will be

We see that CO = CA = CB

Hence C is equidistant from the vertices O, A and B.

Hence proved.

Hope given RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.4 are helpful to complete your math homework.

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