In worksheet on area of polygon we will find the area of triangle, quadrilateral, pentagon etc,. using the formula of area of the triangle formed by three co-ordinate points.
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1. A (-1, 5), B (3, 1) and C (5, 7) are the vertices of the ∆ ABC. If D, E and F are the mid-points of the sides BC, CA and AD respectively, find the area of the ∆ DEF. Show also that, ∆ ABC = 4 ∆ DEF.
2. The co-ordinates of A, B, C are (6, 3), (-3, 5) and (4, -2) respectively and P is the point (x, y) ; show that,
(area of the ∆ PBC)/(area of the ∆ ABC) = |(x + y - 2)/7|
3. The vertices A, B and C of the ∆ABC have co-ordinates (-3, -2,), (2, -2) and (6, 1) respectively. Find the area of the ∆ ABC and the length of the perpendicular from A on BC.
4. It the points A and B have co-ordinates (a cos θ, b sin θ) and (-a sinθ, b cos θ) respectively and O is the origin, then show that the area of the ∆ OAB is independent of θ.
5. The points P, Q, R are collinear; if the co-ordinates of P and Q be (3, 4)and (7, 7) respectively and PR = 10 units, find the co-ordinates of R.
6. The co-ordinates of the points A, B, C, D are respectively (6, 3), (-3, 5), (4, -2) and (x, 3x); if (area of the ∆ PBC)/(area of the ∆ ABC)= 1/2, find x.
7. The co-ordinates of the points A, B, C and D are (-2, 3), (8, 9), (0, 4) and (3, 0) respectively. Find the ratio in which the line-segment AB is divided by the segment, CD.
8. The co-ordinates of the points A and B are (3, 4) and (5, -2) respectively; if PA = PB and the area of the ∆ PAR = 10 sq. units, find the co-ordinates of P.
9. Find the area of the quadrilateral whose vertices have co-ordinates:
(i) (1, 1), (3, 4), (5, -2) and (1, -7).
(ii) (1, 4), (-2, 1), (-2, 3) and (3, 3).
10. The co-ordinates of the vertices A, B, C and D of the quadrilateral ABCD are (1, 2), (-5, 6),
(7, -4) and (k, -2); if the area of the quadrilateral be zero, then find the value of k.
11. The area of a quadrilateral is 28 sq. units. If the co-ordinates of its angular points be (-1, 6), (-2, -4), (3, -2) and (a, b), then show that, 2a + b = 6 or, 2a + b + 22 = 0.
12. Show that the area of the quadrilateral whose vertices taken in order are (a, 0), (-b, 0), (0, a) and (0, -b) is zero (a > 0, b> 0). Give the geometrical significance of the result.
13. Find the area of the pentagon whose vertices have co-ordinates
(0, 1), (2, -3), (5, -4), (4, 0) and (3, 2).
Answers for the worksheet on area of polygon are given below to check the exact answers of the above questions.
1. 4 sq units
3. 7.5 sq. units, 3 units
5. (11, 10) or, (- 5, - 2)
6. 11/8
7. 11 : 47
8. (7, 2) or , (1, 0) ;
9. (i) 20.5 sq. units
(ii) 18.5 sq. units ;
10. K = 3.
13. 16 sq. units
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