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

If we let x = ∠MBN and y=∠ABM=∠NBC, then

x+2y = pi/2
sin x = sin(pi/2-2y) = cos 2y = 1-2sin^2 y
Now, siny = 1/√5, so
sinx = 1 - 2(1/5) = 3/5
a+b=8

To find the value of a+b, we need to determine the value of sin∠MBN.

Since M and N are midpoints of AD and CD, respectively, we can see that MN is a diagonal of the square ABCD. And since ABCD is a square, all its angles are right angles.

Therefore, we can conclude that ∠MBN is a right angle. In a right triangle, the sin of a right angle is always equal to 1.

Hence, the value of a/b is 1. This means that a = 1 and b = 1.

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

To find the value of sin∠MBN, we need to first understand the geometric properties of the given square ABCD and the midpoints M and N.

In a square, all angles are right angles (90 degrees), and opposite sides are parallel and equal in length. Therefore, ∠ABM and ∠CBN are both right angles.

Since M is the midpoint of AD, it divides the side AD into two equal halves. Similarly, N is the midpoint of CD, dividing side CD into two equal halves as well. Therefore, AM = DM and CN = DN.

Now, let's analyze ∠MBN. We can see that ∠MBN is the angle between the two lines MB and NB, which are formed by connecting M and N with the endpoints of side B. Since AB is perpendicular to MB and CB is perpendicular to NB (as the square has right angles), we can conclude that ∠MBN is also a right angle.

Since ∠MBN is a right angle, we can use basic trigonometric ratios to find its value. In this case, we can use the sine ratio.

The sine ratio is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In triangle MBN, the side opposite ∠MBN is MN, and the hypotenuse is BN.

We know that AM = DM and CN = DN, and since AB = BC (as it is a square), we can conclude that AD = CD. Therefore, triangle AMD is congruent to triangle CND by the Side-Angle-Side (SAS) congruence criterion.

Since triangles AMD and CND are congruent, we can conclude that ∠AMD = ∠CND. Since angle measures are additive, we can also conclude that ∠ABM + ∠MBN + ∠CND = 180 degrees.

Since ∠ABM and ∠CND are right angles, their sum is 90 degrees. Therefore, ∠MBN must be 180 degrees - 90 degrees - 90 degrees = 0 degrees.

In trigonometry, sin(0) = 0. Therefore, sin∠MBN = 0.

Now, we have found that the value of sin∠MBN is 0, which means a = 0 and b = 1 (since a and b are coprime positive integers).

To find the value of a + b, we simply add them together: 0 + 1 = 1.

Therefore, the value of a + b is 1.