the perimeter of a rhombus is 36 inches. if the longer diagonal is 14 inches what is the length of the shorter diagonal of the rhombus?
so each side is 9 inches.
A property of a rhombus is that its diagonals right-bisect each other
Let the other diagonal be 2y inches, so we have
4 congruent right-angled triangles with sides 7 and y and hypotenuse of 9
y^2 + 7^2 = 9^2
y^2 = 32
y = √32 = 4√2
so the other diagonal is 8√2
To find the length of the shorter diagonal of a rhombus, we can use the properties of a rhombus.
A rhombus is a quadrilateral with all sides of equal length. The diagonals of a rhombus bisect each other at right angles.
In this problem, the perimeter of the rhombus is given as 36 inches. Since all sides of a rhombus are equal, each side of the rhombus is 36 inches divided by 4, which gives us 9 inches.
We are also given that the longer diagonal is 14 inches. Let's call the longer diagonal "d1" and the shorter diagonal "d2".
Since the diagonals bisect each other at right angles, they divide the rhombus into four congruent right triangles.
Using the Pythagorean theorem, we can find the length of the other side of the right triangle, which corresponds to half of the shorter diagonal of the rhombus.
The Pythagorean theorem states that for a right triangle, the square of the length of the hypotenuse (in this case, d1) is equal to the sum of the squares of the other two sides.
In our case, (d2/2)^2 + (9/2)^2 = d1^2.
Simplifying, (d2^2 + 81)/4 = 196.
Multiplying both sides by 4 gives us d2^2 + 81 = 784.
Subtracting 81 from both sides gives us d2^2 = 703.
Finally, taking the square root of both sides gives us d2 ≈ 26.51 inches.
So, the length of the shorter diagonal of the rhombus is approximately 26.51 inches.