Question

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.

Answer 1

`\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`