In the worksheet on Cartesian triangle the questions are based on proving different types of properties of triangles using coordinate 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 midpoints of the sides of the given triangle.
5. Using analytical method prove that the midpoint of the hypotenuse of a rightangled triangle is equidistant from the three vertices.
6. XYZ is a rightangled triangle, rightangled at Y. If M and N are the midpoints 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 coordinate 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 midpoints of opposite sides of a quadrilateral and the line joining the midpoints of its diagonals meet in a point and bisect one another.
● Coordinate Geometry
11 and 12 Grade Math
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