SOLUTION: A rhombus has sides of length 7 cm. One of its diagonals is 10 cm long. Find the lengths of the other diagonal.

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Question 95565: A rhombus has sides of length 7 cm. One of its diagonals is 10 cm long. Find the lengths of the other diagonal.
Found 2 solutions by Edwin McCravy, ankor@dixie-net.com:
Answer by Edwin McCravy(6938) About Me  (Show Source):
You can put this solution on YOUR website!
A rhombus has sides of length 7 cm. One of its diagonals is 10 cm long. Find the length of the other diagonal.


drawing%28300%2C280%2C-6%2C6%2C-6%2C6%2C%0D%0A+++%0D%0A+++line%28-5%2C0%2C0%2Csqrt%2824%29%29%2C+line%285%2C0%2C0%2Csqrt%2824%29%29%2C%0D%0A+++line%28-5%2C0%2C0%2C-sqrt%2824%29%29%2C+line%285%2C0%2C0%2C-sqrt%2824%29%29%2C%0D%0A+++locate%283%2C3%2C7%29%2C+locate%28-3%2C3%2C7%29%2C+locate%28-3.2%2C-2%2C7%29%2C+locate%283.2%2C-2%2C7%29%0D%0A+%29

As you can see from the drawing, it looks almost like 
the rhombus is a square, so the other diagonal should 
be close to 10 also. Let's suppose the horizonral 
diagonal is the one that is 10 cm. Let's draw it in:

drawing%28300%2C280%2C-6%2C6%2C-6%2C6%2C+line%28-5%2C0%2C5%2C0%29%2C%0D%0A+++%0D%0A+++line%28-5%2C0%2C0%2Csqrt%2824%29%29%2C+line%285%2C0%2C0%2Csqrt%2824%29%29%2C%0D%0A+++line%28-5%2C0%2C0%2C-sqrt%2824%29%29%2C+line%285%2C0%2C0%2C-sqrt%2824%29%29%2C%0D%0A+++locate%283%2C3%2C7%29%2C+locate%28-3%2C3%2C7%29%2C+locate%28-3.2%2C-2%2C7%29%2C+locate%283.2%2C-2%2C7%29%2C%0D%0A+++locate%28-.3%2C-.3%2C10%29%0D%0A+%29   

We want to find the other diagonal, so let's draw it in:

drawing%28300%2C280%2C-6%2C6%2C-6%2C6%2C+line%28-5%2C0%2C5%2C0%29%2C%0D%0A+++line%280%2Csqrt%2824%29%2C0%2C-sqrt%2824%29%29%2C%0D%0A+++line%28-5%2C0%2C0%2Csqrt%2824%29%29%2C+line%285%2C0%2C0%2Csqrt%2824%29%29%2C%0D%0A+++line%28-5%2C0%2C0%2C-sqrt%2824%29%29%2C+line%285%2C0%2C0%2C-sqrt%2824%29%29%2C%0D%0A+++locate%283%2C3%2C7%29%2C+locate%28-3%2C3%2C7%29%2C+locate%28-3.2%2C-2%2C7%29%2C+locate%283.2%2C-2%2C7%29%2C%0D%0A+++locate%28-.3%2C-.3%2C10%29%0D%0A+%29 

Notice that this vertical diagonal divides the 10 cm diagonal
into two equal segments which are 5 cm each. In fact the two
diagonals have divided the rhombus into four congruent right
triangles:

drawing%28300%2C280%2C-6%2C6%2C-6%2C6%2C+line%28-5%2C0%2C5%2C0%29%2C%0D%0A+++line%280%2Csqrt%2824%29%2C0%2C-sqrt%2824%29%29%2C%0D%0A+++line%28-5%2C0%2C0%2Csqrt%2824%29%29%2C+line%285%2C0%2C0%2Csqrt%2824%29%29%2C%0D%0A+++line%28-5%2C0%2C0%2C-sqrt%2824%29%29%2C+line%285%2C0%2C0%2C-sqrt%2824%29%29%2C%0D%0A+++locate%283%2C3%2C7%29%2C+locate%28-3%2C3%2C7%29%2C+locate%28-3.2%2C-2%2C7%29%2C+locate%283.2%2C-2%2C7%29%2C%0D%0A+++locate%28-2%2C-.3%2C5%29%2C+locate%282%2C-.3%2C5%29%0D%0A+%29

Now we pick only one of the four congruent right 
triangles, say, we pick the upper left one:

drawing%28300%2C280%2C-6%2C6%2C-6%2C6%2C+line%28-5%2C0%2C0%2C0%29%2C%0D%0A+++line%280%2Csqrt%2824%29%2C0%2C0%29%2C%0D%0A+++line%28-5%2C0%2C0%2Csqrt%2824%29%29%2C+%0D%0A+++%0D%0A+++locate%28-3%2C3%2C7%29%2C+%0D%0A+++locate%28-2%2C-.3%2C5%29%0D%0A+%29

The hypotenuse, call it c, is 7, and the horizontal
(bottom) leg a is 5, and we need to find the vertical
side (the right side), which we call b:

drawing%28300%2C280%2C-6%2C6%2C-6%2C6%2C+line%28-5%2C0%2C0%2C0%29%2C%0D%0A+++line%280%2Csqrt%2824%29%2C0%2C0%29%2C%0D%0A+++line%28-5%2C0%2C0%2Csqrt%2824%29%29%2C+%0D%0A+++locate%28.3%2C2%2C%27b=%3F%27%29%2C%0D%0A+++locate%28-3.5%2C3%2C%27c=7%27%29%2C+%0D%0A+++locate%28-2%2C-.3%2C%27a=5%27%29+%0D%0A+%29

so now we use the Pythagorean theorem to find the 
leg b (the right side).

      c² = a² + b²

      7² = 5² + b²

      49 = 25 + b²
Subtract 25 from both sides:

 49 - 25 = b²

      24 = b²
      __
     Ö24 = b
                          _
(This can be written as 2Ö6 if you have
 learned how to get radicals to lowest terms.)

Or it can be approximated with the decimal
       b = 4.898979486 cm.

But the diagonal of the rhombus is twice this
or 
  __      _
2Ö24 or 4Ö6 or 9.797958971 cm.  So this is very
close to 10 cm, so now we know why it looks so
much like a square.  It is very nearly a square.

Edwin


Answer by ankor@dixie-net.com(12701) About Me  (Show Source):
You can put this solution on YOUR website!
A rhombus has sides of length 7 cm. One of its diagonals is 10 cm long. Find the lengths of the other diagonal.
:
We know that the two diagonals bisect each other at right angles.
This forms a right triangle with a side of 5 cm and hypotenuse of 7 cm
:
1. Find one of the angles(A) of the right triangle using the cosine; 5%2F7
2. Then find the 3rd side using the sine of this same angle.
:
3. Double the value of that side and you have the length of the other diagonal.
:
Cosine(A) = 5%2F7
A = 44.415 degrees
:
Sin(44.4153) = a%2F7
.69985 = a%2F7
a = 7 * .69985
a = 4.899 * 2 = 9.8 cm is the length of the other diagonal
:
Did this make sense to you?