the perimeter of a rhombus is 68 one diagonal is 30 find the length of the other diagonal

Question

the perimeter of a rhombus is 68 one diagonal is 30 find the length of the other diagonal

Answer 1

By definition, a rhombus is a quadrilateral made up of 4 equal sides.  With this information, we can figure out that the length of each of the sides is as follows:

 

P= 4 * S or 68=4 * S

S= 17

 

When dealing with diagonals in a rhombus, the diagonals splits the rhombus into four right triangles.  We can use this to find the length of the unknown diagonal.  Looking at the triangle, the length given by the diagonal is actually cut in half to 15.  So we have a triangle that looks like this:

.                         x=15 

. .                       y=17

.   .                     z=?

.     .

.       .

x        y

.          .

.           .

....z.........

 

We need to find "z" which is half of our missing diagonal.  To solve this, we use Pythagorean Theorem:

 

x² + z² = y²

 

We can rearrange this to solve for "z"

 

z = (y² - x²)^1/2

 

Plug in our numbers we get

 

z = (17² - 15²)^1/2 = (289 - 225)^1/2 = (64)^1/2 = 8

 

So "z" is equal to 8.  Now we are not down yet, as this is only half of the new diagonal.  So we double it, which is 16.

 

Length of the other diagonal is: 16

 

 

Answer 2

The perimeter of the rhombus is 68.  We know then that each side of the rhombus is 68/4 = 17.

 

Now, the diagonals of the rhombus bisect each other at right angles.  We then have 4 right triangles, each with one leg = 30/2  = 15 and the hypotenuse = 17.

 

Using the pythagorean theorem, we know that for any right triangle:

 

a² + b² = c²

 

Solving this equation for the missing leg, we get:

 

15² + b² = 17²

 

225 + b² = 289

 

b² = 64

 

b = 8

 

So, half the other diagonal is 8 => the second diagonal's length is 16.