ABCD is a square where M and N are midpoints of AD and CD , respectively. If sin�ÚMBN=a b , where a and b are coprime positive integers, what is the value of a+b ?

Well, well! A square, M, N, and some sin�ÚMBN action going on. Looks like we're in for some geometric fun!

Since M is the midpoint of AD and N is the midpoint of CD, we can conclude that MN is parallel to AC and that MN is also equal in length to AC. This makes MN the diagonal of the square.

Now, let's look at triangle MBN. Since it's a right triangle (because M and N are midpoints and sides of a square are perpendicular), we can use Pythagoras' theorem.

Let's say the side length of the square is s. Then, the length of MN (or the diagonal) is s√2.

Applying Pythagoras' theorem, we have:

(√2s)^2 = BN^2 + NM^2
2s^2 = BN^2 + (s/2)^2
2s^2 = BN^2 + s^2/4

Now, let's think about sin�ÚMBN. By definition, it's the length of the side opposite the angle divided by the length of the hypotenuse.

So, sin�ÚMBN = BN / MN

Substituting the values we found earlier, sin�ÚMBN = BN / (s√2)

From our earlier equation, we had 2s^2 = BN^2 + s^2/4. Rearranging it, we have BN = √(7/8) * s.

Substituting that back into our sin�ÚMBN equation, we get sin�ÚMBN = √(7/8) * s / (s√2)
= √7 / (2√2)
= √7 / (2√2) * √2 / √2
= √14 / 4

So, a = √14 and b = 4. Since they are coprime positive integers, the value of a+b is 14 + 4 = 18.

And just like that, we've solved the puzzle! Keep those geometric brain teasers coming!

To find the value of a + b, we need to first determine the value of sin(�ÚMBN).

Let's start by drawing the square ABCD and adding the points M and N:

A ------- B
| |
| |
| M |
| |
| |
D ------- C N

Since M is the midpoint of AD, we can see that MB is equal in length and parallel to AB. Similarly, since N is the midpoint of CD, we can see that BN is equal in length and parallel to BC.

Looking at triangle MBN, we can observe that MB and BN are equal in length, making the triangle isosceles. Additionally, MN is equal in length to half the length of AD (which is equal to half the length of BC since ABCD is a square).

Let x represent the length of MB (and BN) and y represent the length of AD (and BC).

Since triangle MBN is isosceles, we can draw a height from M to the midpoint of BN, and this height will bisect the base(MB) and be perpendicular to BN.

Now, in right triangle MBQ (where Q is the point of intersection between the height and BN), we have:

MB = x
MQ (altitude) = y/2
BQ (base) = x/2

Using the Pythagorean theorem, we can find the length of MQ:

(MQ)^2 = (MB)^2 - (BQ)^2
(y/2)^2 = x^2 - (x/2)^2
(y/2)^2 = x^2 - x^2/4 (Using (a-b)^2 = a^2 - 2ab + b^2)
(y^2)/4 = 3(x^2)/4
y^2 = 3x^2

Now, considering triangle MBN, we want to find sin(�ÚMBN):

sin(�ÚMBN) = (opposite side) / (hypotenuse)
= MN / BN

Since we know MN = y/2 and BN = x, we can substitute these values:

sin(�ÚMBN) = (y/2) / x
= y / (2x)

Based on our earlier equation (y^2 = 3x^2), we can substitute y with sqrt(3) * x:

sin(�ÚMBN) = (sqrt(3) * x) / (2x)
= sqrt(3) / 2

Therefore, the value of sin(�ÚMBN) is sqrt(3) / 2.

Since sqrt(3) and 2 are coprime, the value of a = sqrt(3) and b = 2. Therefore, a + b = sqrt(3) + 2.

To find the value of a+b, we need to first understand the problem and identify any given information.

We are given that ABCD is a square, and M and N are the midpoints of sides AD and CD, respectively. Now, let's analyze the figure and see what we can determine.

Since ABCD is a square, all sides are equal in length. Let's assume the length of each side is 's'.

We can use this information to find the length of MN. Since M and N are the midpoints of AD and CD, respectively, MN is half the length of AD or CD.

MN = s/2

Now, let's find the length of MB and BN.

MB = AB - AM
BN = DN - ND

Since ABCD is a square, AB = AD = s, and DN = CD = s.

MB = s - (s/2) (from the property of midpoint)
= s/2

BN = s - (s/2) (from the property of midpoint)
= s/2

Now, we have the lengths of the following sides:
MN = s/2
MB = s/2
BN = s/2

To find the value of sin(�ÚMBN), we can use the formula for the sine of an angle in a right triangle. In triangle MBN, we have a right angle at B.

sin(�ÚMBN) = opposite/hypotenuse = MB/MN = (s/2)/(s/2) = 1

Therefore, sin(�ÚMBN) = 1.

Since the coprime positive integers a and b are not given directly, and we are only required to find the value of a+b, it is clear that a = 1 and b = 1.

Therefore, the value of a+b = 1 + 1 = 2.