**Q) Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So he started to sketch his own rocket designs on the graph sheet. One such design is given below :**

**Based on the above, answer the following questions : **

**(i) Find the mid-point of the segment joining F and G.**

**(ii) What is the distance between the points A and C?**

**(iii) Find the coordinates of the point which divides the line segment joining the points A and B in the ratio 1 : 3 internally.**

**(iv) What are the coordinates of the point D?**

**Ans: **

(i) M**id-point of the segment joining F and G:**

We know that the coordinates of midpoint are given by:

(X,Y) =

From the diagram, we have coordinates of F (- 3, 0) and G (1, 4)

By substituting the given values, we get:

(x , y) =

∴ (x , y) = (-1, 2)

**Therefore, coordinates of the midpoints of the line joining F & G is (- 1, 2).**

**(ii) Distance between the points A and C:**

We know that the distance between two points P (X_{1}, Y_{1}) and Q (X_{2}, Y_{2}) is given by:

PQ =

From the diagram, we have co-ordinates as A (3, 4) and C (- 1, – 2)

∴ Distance AC =

∴ AC =

∴ AC = √52 = 2 √13

**Therefore, the distance between AC is 2 √13 units.**

**(iii) Coordinates of the point dividing line AB in 1 : 3:**

By section formula, coordinates of point P (X, Y) which lies between two points (x_{1}, y_{1}), (x_{2}, y_{2}) will be given by:

P (X,Y) =

From the given diagram, we have coordinates of points A (3, 4) and B (3, 2)

Since, the point divides the line in ratio of 1:3, therefore m1 = 1 and m2 = 3

P (X,Y) =

=

=

**Therefore, the coordinates of the point is which divides line AB in ratio of 1:3.**

**(iv) Coordinates of the point D:**

From the diagram, coordinates of the point D are (-2, -5).

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