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