Worksheet on Area of Polygon

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.

Answers:

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

● Co-ordinate Geometry 

• What is Co-ordinate Geometry?
  
• Rectangular Cartesian Co-ordinates
  
• Polar Co-ordinates
  
• Relation between Cartesian and Polar Co-Ordinates
  
• Distance between Two given Points
  
• Distance between Two Points in Polar Co-ordinates
  
• Division of Line Segment: Internal & External
  
• Area of the Triangle Formed by Three co-ordinate Points
  
• Condition of Collinearity of Three Points
  
• Medians of a Triangle are Concurrent
  
• Apollonius' Theorem
  
• Quadrilateral form a Parallelogram 
  
• Problems on Distance Between Two Points 
  
• Area of a Triangle Given 3 Points
  
• Worksheet on Quadrants
  
• Worksheet on Rectangular – Polar Conversion
  
• Worksheet on Line-Segment Joining the Points
  
• Worksheet on Distance Between Two Points
  
• Worksheet on Distance Between the Polar Co-ordinates
  
• Worksheet on Finding Mid-Point
  
• Worksheet on Division of Line-Segment
  
• Worksheet on Centroid of a Triangle
  
• Worksheet on Area of Co-ordinate Triangle
  
• Worksheet on Collinear Triangle
  
• Worksheet on Area of Polygon
  
• Worksheet on Cartesian Triangle

11 and 12 Grade Math 

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