Long Answer TypeYou have studied in Class IX, (Chapter 9, Example 3), that a median of a triangle divides it into two triangles of equal areas. Verify this result for Δ ABC whose vertices are A(4, – 6), B(3, –2) and C(5, 2)
Short Answer TypeDetermine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).
Find the centre of a circle passing through the points (6, – 6), (3, – 7) and (3, 3).
Long Answer TypeThe two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices.
Fig. 7.18.
Then, AB = BC
⇒ AB2 = BC2
⇒ (x + 1)2 + (y - 2)2
= (x - 3)2 + (y - 2)2
⇒ x2 + 2x + 1 + y2 - 4y + 4
= x2 - 6x + 9 + y2 - 4y + 4
⇒ 8x = 8
⇒ x = 1
Also,AB2 + BC2 = AC2
[∵ ∠B = 90° and therefore using Pythagoras theorem]
⇒ (x + 1)2 + (y - 2)2 + (x - 3)2 + (y - 2)2
= (3 + 1)2 + (2 - 2)2
⇒ x2 + 2x + 1 + y2 - 4y + 4 + x2 - 6x + 9 + y2 - 4y + 4 = 16
⇒ 2x2 + 2y2 - 4x - 8y + 2 = 0
⇒ x2 + y2 - 2x - 4y + 1 = 0
[Dividing throughout by 2]
Putting x = 1, we get
⇒ 1 + y2 - 2 - 4y + 1 = 0
⇒ y(y - 4) = 0
⇒ y = 0,4
Hence, the other vertices are (1, 0) and (1, 4).
The Class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of 1m from each other. There is a triangular grassy lawn in the plot as shown in the Fig. 7.14. The students are to sow seeds of flowering plants 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?
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 the ∆ADE and compare it with the area of ∆ABC.
Short Answer TypeLet A (4, 2), B(6, 5) and C(1, 4) be the vertices of Δ ABC.
The median from A meets BC at D. Find the coordinates of the point D.
Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of Δ ABC.
Find the coordinates of the point P on AD such that AP : PD = 2 : 1
Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of Δ ABC.
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.
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