Worksheet on Cartesian Triangle

In the worksheet on Cartesian triangle the questions are based on proving different types of properties of triangles using co-ordinate geometry.

1. If G be the centroid of the triangle XYZ, then prove that, YZ² + ZX² + XY² = 3(GX² + GY² + GZ²)

Area of the ∆ XYZ = 3 × area of the ∆GYZ. 

2. Show that the straight line joining the middle points of two sides of a triangle is equal to half the third side. 

3. Show that the straight lines joining the middle points of the opposite sides of a quadrilateral bisect each other. 

4. Prove analytically that the area of a triangle is four times that of the triangle obtained by joining the mid-points of the sides of the given triangle. 

5. Using analytical method prove that the mid-point of the hypotenuse of a right-angled triangle is equidistant from the three vertices. 

6. XYZ is a right-angled triangle, right-angled at Y. If M and N are the mid-points of the sides XY and YZ respectively, then show that,
4(XN² + ZM²) = 5 ∙ XZ².

7. Prove analytically that the sum of squares of the three sides of a triangle is equal to four times the sum of squares of its medians.

8. Using co-ordinate geometry prove that an isosceles triangle has two equal medians.

9. If two medians of a triangle are equal, prove analytically that the triangle is isosceles.

10. Prove analytically that the lines joining the mid-points of opposite sides of a quadrilateral and the line joining the mid-points of its diagonals meet in a point and bisect one another.

● 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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