Q8 of 33 Page 171

ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q,R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateralPQRS a square? a rectangle? or a rhombus? Justify your answer

P is the mid-point of AB


Coordinates of P = (-1, )


Similarly coordinates of Q, R and S are (2, 4), (5, ) and (2, -1) respectively


Length of PQ = [(-1 – 2)2 + ()2]1/2


=


Length of QR = [(2 – 5)2 + ()2]1/2


=


Length of RS = [(5 – 2)2 + (+ 1)2]1/2


=


Length of SP = [(2+ 1)2 + (-1 – )2]1/2


=


Length of PR = [(-1- 5)2 + (– )2]1/2


= 6


Length of QS = [(2 – 2)2 + (4 + 1)2]1/2


= 5


It can be observed that all sides of the given quadrilateral are of the same measure. However, the diagonals are ofdifferent lengths. Therefore, PQRS is a rhombus


More from this chapter

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The two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices

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The Class X students of asecondary school inKrishinagarhave been allotteda rectangular plot of land fortheir gardening activity.Sapling of Gulmohar areplanted on the boundary at adistance of 1m from each other.There is a triangular grassylawn in the plot as shown inthe Fig. 7.14. The students areto sow seeds of floweringplants on the remaining area of the plot

(i) Taking A as origin, find the coordinates of the vertices of the triangle.


(ii) What will be the coordinates of the vertices of Δ PQR if C is the origin?


Also calculate the areas of the triangles in these cases. What do you observe?


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The vertices of a Δ ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that Calculate the area of theΔ ADE and compare it with the area of Δ ABC (Recall Theorem 6.2 and Theorem 6.6)

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Let A (4, 2), B(6, 5) and C (1, 4) be the vertices of Δ ABC

(i) The median from A meets BC at D. Find the coordinates of the point D


(ii) Find the coordinates of the point P on AD such that AP: PD = 2: 1


(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ: QE = 2: 1 and CR: RF = 2: 1


(iv) What do you observe?


[Note: The point which is common to all the three medians is called the centroidand this point divides each median in the ratio 2: 1]


(v) If and are the vertices of Δ ABC, find the coordinates of the centroid of the triangle