**Q) A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. ****In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure.**

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

**(i) If A is taken as origin, what are the coordinates of the vertices of Δ PQR? **

**(ii) Find distances PQ and QR. **

**(iii) Find the coordinates of the point which divides the line segment joining points P and R in the ratio 2 : 1 internally. **

**(iv)** **Find out if D PQR is an isosceles triangle. **

**Ans:**

**(i) coordinates of the vertices of Δ PQR:**

from the diagram, if we take A as origin, we get following:

**Coordinates of point P: (4, 6)**

**Coordinates of point Q: (3, 2)**

**Coordinates of point R: (6, 5)**

**(ii) Distances PQ and QR:**

We know that the distance between two points P (X1, Y1) and Q (X2, Y2) is given by:

PQ =

From the diagram, we have co-ordinates as P (4, 6) and Q (3, 2)

∴ Distance PQ =

∴ PQ =

∴ PQ = √17

**∴ PQ = √17 units**

Similarly, from the diagram, we have co-ordinates as Q (3, 2) and R (6, 5)

∴ Distance QR =

∴ QR =

∴ QR = √18

**∴ QR = 2√3 units**

**Therefore, the length of PQ is √17 units and QR is 2√3 units.**

**(iii) coordinates of the point dividing the line PR in the ratio 2 : 1 internally:**

By section formula, coordinates of point P (X, Y) which lies between two points (x1, y1), (x2, y2) will be given by:

P (X,Y) =

From the given diagram, we have coordinates of points P (4, 6) and R (6, 5)

Since, the point divides the line in ratio of 2:1, therefore m1 = 2 and m2 = 1

P (X,Y) =

**= **

**Therefore, the coordinates of the point is which divides line AB in ratio of 2:1.**

**(iv) Is Δ PQR an isosceles triangle?:**

If Δ PQR is an isosceles triangle, then its two sides will be equal.

Coordinates of all 3 vertices are: P(4, 6), Q(3, 2) and R(6, 5)

Therefore, let’s check length of all the 3 sides:

We know that the distance between two points P (X1, Y1) and Q (X2, Y2) is given by:

**PQ = √17 units** (from part (ii) above)

**QR = 2√3 units** (from part (ii) above)

PR =

∴ PR =

**∴ PR = √5 units**

**Since all 3 sides are unequal, therefore Δ PQR is NOT an isosceles triangle.**

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