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What is the area of a rhombus of side 13 such that the sum of its two diagonals is 34?

 

 May 15, 2020
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A formula for finding the area of a rhombus, given the two diagonals d1 and d2, is:  Area  =  ½·d1·d2

 

Since the diagonals of a rhombus are perpendicular, they form four right triangles.

 

Since the sum of its two diagonals is 34, the sum of one-half of one diagonal (one side of a right triangle)

and one-half of the other diagonal (the other side of a right triangle) is 17.

 

Let one side be  x  then the other side is  17 - x.

 

Using the Pythagorean Theorem:  (x)2 + (17 - x)2  =  132

                                                x2 + 289 - 34x + x2  =  169

                                                      2x2 - 34x + 120  =  0

                                                          x2 - 17x + 60  =  0

                                                          (x - 5)(x - 12)  =  0

                                                                      x = 5  or  x = 12

 

So, the diagonals are  10  and  24  and the area is  ½·10·24  =  120

 May 15, 2020

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