Square ABCD has M as the midpoint of AB, N as the midpoint of BC, P as the midpoint of CD and Q as the midpoint of MP. If [AMNPDQ]=20, what is the value of [ABCD]?
Details and assumptions
[PQRS] refers to the area of figure PQRS.
To find the area of square ABCD, we need to know the lengths of its sides. Let's assign a variable to the length of one side of the square.
Let x be the length of one side of square ABCD.
Since M is the midpoint of AB, we can deduce that AM = MB = x/2.
Similarly, since N is the midpoint of BC, we have BN = NC = x/2.
Since P is the midpoint of CD, we have CP = PD = x/2.
Now, let's consider quadrilateral AMNQ.
We know that AM = x/2 and there is a diagonal MP.
Let's determine the length of MP using the Pythagorean theorem:
MP^2 = AM^2 + AP^2
Since AM = x/2 and AP = CP + PA = x/2 + x/2 = x, we have:
MP^2 = (x/2)^2 + (x)^2
MP^2 = (x^2)/4 + x^2
MP^2 = (5x^2)/4
Next, we need to find the length of AQ. Since Q is the midpoint of MP, we have:
AQ = 2MQ
Since MP = x/2, we have:
AQ = 2 * (x/2) = x
Now, we have a right-angled triangle AMQ, with AM = x/2, AQ = x, and MP = x/2.
The area of triangle AMQ can be found using the formula:
[AMQ] = 0.5 * base * height
[AMQ] = 0.5 * (x/2) * x
[AMQ] = (x^2)/4
Since AQ = x, quadrilateral AMNQ can be divided into two triangles of equal areas: AMQ and ANQ.
Therefore, the area of quadrilateral AMNQ is:
[AMNQ] = 2 * [AMQ] = 2 * ((x^2)/4) = (x^2)/2
Now, let's find the area of triangle NPQ.
Since N is the midpoint of BC, we have:
QN = (1/2) * NC = (1/2) * (x/2) = x/4
Similarly, since P is the midpoint of CD, we have:
PQ = (1/2) * PD = (1/2) * (x/2) = x/4
The base of triangle NPQ is PQ = x/4, and its height is AQ = x.
So the area of triangle NPQ is:
[NPQ] = 0.5 * (base) * (height)
[NPQ] = 0.5 * (x/4) * x
[NPQ] = (x^2)/8
Lastly, to find the area of square ABCD, we can subtract the sum of the two triangles from the original area [AMNPDQ]:
[ABCD] = [AMNPDQ] - [AMNQ] - [NPQ]
[ABCD] = 20 - (x^2)/2 - (x^2)/8
[ABCD] = 20 - ((4x^2)/8) - ((x^2)/8)
[ABCD] = 20 - (5x^2)/8
So, the value of [ABCD] is (20 - (5x^2)/8).