Question 1078568: ABCD is a rhombus with perimeter 120 units and area 720 square units. P is the midpoint of line segment AD. find the area and perimeter of triangle ACP
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! It would be good to know the side length of this rhombus.
A rhombus has 4 sides of the same length,
so a rhombus with perimeter 120 units,
has a side length of  .
A rhombus id a parallelogram,
so if we take one of its sides as the base,
its area (in square units) would be
 ,
where  is the height in units.
So,  .
It would also be good to understand what we need to find out,
so we need a sketch.
Since  is the midpoint of AD,
obviously  .
AREA OF ACP:
Any way you look at it, the area of ACP is
 of the area of ABCD,
so (in square units) it is  .
1) Either you realize that
the area of ACD is  of the area of ABCD,
and when you split ACD in equal parts with median PC,
the areas of the halves (ACP and DCP) are of the area od ACD.
2) Or you realize that the area of parallelogram ABCD is  ,
while the area of triangle ACD is,
with  .
PERIMETER OF ACP:
We know  .
We just need  and  .
The Pythagorean theorem, applied to right triangle DCX, tells us that
 ,
or with all lengths in length units,
 -->  -->  .
With that, with all length in length units, we find
 , and  .
The, applying the Pythagorean theorem to right triangles ACX and PCX,
with all lengths in length units,
 -->  , and
 -->  .
So the perimeter of ACP is
  units.
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