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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