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In square ABCD, M is the midpoint of AB, and N is the midpoint of CD. If the side length of the square is 1, then find the area of the shaded region.

In square ABCD, M is the midpoint of AB, and N is the midpoint of CD. If the side-example-1
User Gdiazc
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1 Answer

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\angle ADP=15^\circ

Step-by-step explanation:

As shown in the figure
ABCD is a square with side length
= x, given
M and
N are the midpoints of
AB and
CD, respectively,


\implies DN=(1)/(2) x

if the paper is folded along
DP,
A lands on
Q on segment
MN,


\implies AP=PQ


\implies DQ=AD=x


\Delta DAP and
\Delta DQP are congruent

and
\text{cos} \ \alpha =(DN)/(DQ) =((1)/(2)x )/(x) =(1)/(2)


\implies\alpha =\text{cos}^(-1)\huge \text((1)/(2)\huge \text )=60^\circ


\implies\angle ADQ=2\beta =90-60=30^\circ


\implies \angle ADP=\beta =(1)/(2) *30=15^\circ

In square ABCD, M is the midpoint of AB, and N is the midpoint of CD. If the side-example-1
User Francesco Vadicamo
by
8.7k points

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