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).
Fig. 7.17.
Let the centre of the circle be O(x, y)
Then, OA = OB = OC
[radii of circle]
⇒ (OA)2 = (OB)2 = (OC)2 ....(i)
From (i) we have,
OA2 = OB2
⇒ (x - 6)2 + (y + 6)2
= (x - 3)2 + (y + 7)2
(x - 6)2 + (y + 6)2 = (x - 3)2 + (y + 7)2
⇒x2 - 12x + 36 + y2 + 12y + 36
= x2 — 6x + 9 + y2 + 14y + 49
⇒ 6x + 2y = 14
⇒ 3x + y = 7 ...(i)
Again we have,
(OB)2 = (OC)2
⇒ (y + 7)2 = (y - 3)2
⇒ y2 + 49 + 14y = y2 + 9 - 6y
⇒ 20y = 9 - 49
⇒ 20y = 40
⇒ y = -2 ...(ii)
Putting the value of (iii) in (ii) we get, x = 3
Hence, centre of a circle = 0(3, -2)
Long Answer TypeThe two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices.
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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