Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
In ABCD, let d1 and d2 be the diagonals AC and BD, respectively. Construct a coordinate system so that A is the origin, B lies on the positive x axis, and C and D lies above the x axis. See the diagram below. Assume that AB is equal to DC and AD is equal to BC. We must prove that .
Let x be the x coordinate of B, let C have coordinates (x + x0, y) and D have the coordinates (x0, y). If x0 is 0 then the parallelogram is a rectangle.
Firstly, using the distance formula, let's solve for the length of the two diagonals, d1 and d2.
Now let's sum the squares of the diagonals:
Secondly, again using the distance formula, let's solve for the length of the two sides AB and AD.

Finally, we want to sum the squares of the sides and multiply by 2 (4 sides total, 2 of each length).

Thus, we have proven that the sum of the squares of the four sides of a parallelogram is equal to the sum of the squares of the diagonals.
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 10
You have rated this answer /10
- CBSE Sample Papers for Class 10 Mathematics
- R S Aggarwal and V Aggarwal Textbook Solutions for Class 10 Mathematics
- RD Sharma Textbook Solutions for Class 10 Mathematics
- NCERT Textbook Solutions for Class 10 Mathematics
- CBSE Syllabus for Class 10 Mathematics
- CBSE Previous year papers with Solutions Class 10 Mathematics
Browse free questions and answers by Chapters
- 1 Pair of Linear Equations in 2 Variables
- 2 Quadratic Equations
- 3 Arithmetic Progression
- 4 Triangles
- 5 Some Applications of Trigonometry
- 6 Polynomials
- 7 Coordinate Geometry
- 8 Introduction to Trigonometry
- 9 Circles
- 10 Constructions
- 11 Areas Related to Circles
- 12 Surface Areas and Volumes
- 13 Statistics
- 14 Probability
- 15 Real Numbers