SOLUTION: ABCD is a square with side of 10 cm. 3 points Square PQRS is drawn inside the square ABCD such that trapezoids PQBA, QRCB, SRCD and PSDA are equal. If the area of the trapezoid is

Algebra ->  Circles -> SOLUTION: ABCD is a square with side of 10 cm. 3 points Square PQRS is drawn inside the square ABCD such that trapezoids PQBA, QRCB, SRCD and PSDA are equal. If the area of the trapezoid is       Log On


   

Question 1186474: ABCD is a square with side of 10 cm. 3 points Square PQRS is drawn inside the square ABCD such that trapezoids PQBA, QRCB, SRCD and PSDA are equal. If the area of the trapezoid is 16 cm^2, find the height of the trapezium.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Here is a partial diagram of the figure....



The area of trapezoid SRCD is 16:

A+=+h%28%28b%281%29%2Bb%282%29%29%2F2%29

16+=+x%28%2810%2B10-2x%29%2F2%29
16+=+x%2810-x%29
16+=+10x-x%5E2
x%5E2-10x%2B16=0
%28x-2%29%28x-8%29=0

x=2 or x=8

Obviously from the diagram the x we want is x=2.

ANSWER: the height of the trapezoid is 2cm.

NOTE: The solution from tutor @ikleyn, using the fact that the four trapezoids are congruent, is much easier than my solution above....

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
ABCD is a square with side of 10 cm. highlight%28cross%283_points%29%29 Square PQRS is drawn inside the square ABCD
such that trapezoids PQBA, QRCB, SRCD and PSDA are equal.
If the area of the trapezoid is 16 cm^2, find the height of the trapezium.
~~~~~~~~~~~~~~~~~~ 


From the area of the square ABCD, subtract four times the area of the trapezoid to get the area of the square PQRS


    the area of the square PQRS = 10^2 - 4*16 = 100 - 64 = 36 cm^2.


Hence, the side of the square PQRS is  sqrt%2836%29 = 6.


Since the trapezoids are equal (congruent), it means that the strip between the squares is of uniform width of  %2810+-+6%29%2F2 cm = 4%2F2 = 2 cm.


The width of the strip is the height of the trapezoids.


ANSWER.  The height of the trapezoids is 2 cm.

Solved.


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These words in the condition of the problem 

    highlight%28cross%283_points%29%29 Square PQRS

perplexed me, when I read this problem today in the morning . . .