SOLUTION: ABCD is an isosceles trapezoid with AB=CD. P is the midpoint of line segment AB, Q is the midpoint of line segment CD, PQ=25 units, AD is 35 units, and the area of the trapezoid AB

Algebra ->  Surface-area -> SOLUTION: ABCD is an isosceles trapezoid with AB=CD. P is the midpoint of line segment AB, Q is the midpoint of line segment CD, PQ=25 units, AD is 35 units, and the area of the trapezoid AB      Log On


   

Question 1078570: ABCD is an isosceles trapezoid with AB=CD. P is the midpoint of line segment AB, Q is the midpoint of line segment CD, PQ=25 units, AD is 35 units, and the area of the trapezoid ABCD is 600 square units. Find the perimeter of ABCD.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
ABCD is an isosceles trapezoid with AB=CD.
That means that sides (bases) AD and BC are parallel,
and that pairs of angles adjacent to the same base
(such as DAB and CDA) are congruent.


Drawing segment PQ does not add to understanding the situation,
but knowing that PQ is the midsegment of the trapezoid,
and knowing the length of PQ is useful information.
We know that the length of the midsegment of the trapezoid
is the average of the length of the bases, so
PQ=%28AD%2BBC%29%2F2 or 25=%2835%2BBC%29%2F2 ---> BC=15
We also know that for this trapezoid
area=%28AD%2BBC%29green%28h%29%2F2 or area=PQ%2Agreen%28h%29 ,
so, with length is (length) units, and areas in square units,
600=25green%28h%29 --> green%28h%29=600%2F25=24 .

Isosceles trapezoid ABCD is split by those green lines into a rectangle,
and two congruent right triangles.
So AX=YD=%2835-25%29%2F2=10 (in our length units, of course).
Applying the Pythagorean theorem to those triangles,
AB=CD=sqrt%2810%5E2%2B24%5E2%29=sqrt%28100%2B576%29=sqrt%28676%29=13 .
So, in our length units, the perimeter of ABCD is
AB%2BBC%2BCD%2BAD=13%2B15%2B13%2B35=highlight%2876%29 .