How do you prove that sum of the squares on the sides of a rhombus is equal to the sum of squares on its diagonals?
see explanation.
Some of the properties of a rhombus :
Similarly,
In
by Pythagorean theorem, we know that
Let the sum of squares of the sides be
Let the sum of squares of the diagonals be
By signing up, you agree to our Terms of Service and Privacy Policy
To prove that the sum of the squares on the sides of a rhombus is equal to the sum of squares on its diagonals, let's denote the side length of the rhombus as ( a ) and the lengths of the diagonals as ( d_1 ) and ( d_2 ).

Sum of squares on the sides: The sum of the squares on the sides is ( 4a^2 ).

Sum of squares on the diagonals: The sum of the squares on the diagonals is ( d_1^2 + d_2^2 ).
To prove the given statement, we can use the properties of a rhombus and the Pythagorean theorem.
Consider half of the rhombus formed by one diagonal and two sides, as shown in the figure below:
AD
/ \
/ \
BC
Using the Pythagorean theorem:
For triangle ABD: [ AB^2 + BD^2 = AD^2 ] [ a^2 + \left(\frac{d_1}{2}\right)^2 = \left(\frac{d_2}{2}\right)^2 ] ...(i)
For triangle ACD: [ AC^2 + CD^2 = AD^2 ] [ a^2 + \left(\frac{d_2}{2}\right)^2 = \left(\frac{d_1}{2}\right)^2 ] ...(ii)
Adding equations (i) and (ii):
[ 2a^2 + \frac{d_1^2}{4} + \frac{d_2^2}{4} = \frac{d_1^2 + d_2^2}{2} ]
Multiplying both sides by 4:
[ 8a^2 + d_1^2 + d_2^2 = 2d_1^2 + 2d_2^2 ] [ 8a^2 = d_1^2 + d_2^2 ]
Thus, the sum of the squares on the sides ( 4a^2 ) is equal to the sum of the squares on its diagonals ( d_1^2 + d_2^2 ).
By signing up, you agree to our Terms of Service and Privacy Policy
You can prove that the sum of the squares on the sides of a rhombus is equal to the sum of squares on its diagonals by using the properties of a rhombus and basic geometric principles. One way to prove this is by using the Pythagorean theorem. Since a rhombus is a special type of parallelogram, its diagonals bisect each other at right angles, dividing it into four congruent right triangles. By applying the Pythagorean theorem to each of these triangles and then summing up the results, you can demonstrate the equality between the sum of the squares on the sides and the sum of squares on the diagonals of the rhombus.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 Two opposite sides of a parallelogram each have a length of #10 #. If one corner of the parallelogram has an angle of #(3 pi)/4 # and the parallelogram's area is #70 #, how long are the other two sides?
 How do you calculate the area of a parallelogram if only two diagonals are given?
 Two rhombuses have sides with lengths of #9 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(5pi)/12 #, what is the difference between the areas of the rhombuses?
 A parallelogram has sides A, B, C, and D. Sides A and B have a length of #18 # and sides C and D have a length of # 9 #. If the angle between sides A and C is #pi/4 #, what is the area of the parallelogram?
 Two rhombuses have sides with lengths of #3 #. If one rhombus has a corner with an angle of #(7pi)/12 # and the other has a corner with an angle of #(pi)/6 #, what is the difference between the areas of the rhombuses?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7