In a pentagon ABCDE, EA = AB = BC = CD = 10cm, (angle)EAB = (angle)ECD = 90(degrees) and (angle)ABC = 135(degrees). Find the length of ED.
If you draw a diagram you would see that if we find EC length, we can find ED, since ECD is a right angled triangle.
To find EC we can draw a perpendicular line to EA from C. The point it cuts AC would be F. Now if you observe properly you would see that,
`EF + BC xx cos45 = EA`
`EF + 10 xx 1/sqrt(2) = 10`
`EF = 10 - 10/sqrt(2)`
`EF = 10(1-1/sqrt(2))`
Also CF is given by,
`CF = AB + BC xx sin45`
`CF = 10 + 10/sqrt(2)`
`CF = 10(1+1/sqrt(2))`
Now the triangle EFC is a right angled triangle,
`EC^2 = EF^2 + CF^2`
We also know from ECD triangle,
`ED^2 = CD^2 + EC^2`
`ED^2 = CD^2 + EF^2 + CF^2`
`ED^2 = 10^2 + (10(1-1/sqrt(2)))^2 + (10(1+1/sqrt(2)))^2`
`ED^2 = 10^2(1+1-2/sqrt(2) + 1/2+1+2/sqrt(2)+1/2)`
`ED^2 = 10^2 (1+1+1/2+1+1/2)`
`ED^2 = 10^2(4)`
`ED^2 = 400`
Therefore, `ED = sqrt(400)`
`ED = 20`
Therefore the length of ED is 20.