Two unequal circles with centres A and B intersect each other at points C and D. The centre B of the smaller circles lies on the circumference of the bigger circle with centre A. If ` angle CMD = x^(@) , ` find in terms of x, the measure of angle DAC .
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Answer

To solve the problem, we need to find the measure of angle DAC in terms of x, given that angle CMD = x degrees. Here is the step-by-step solution:

### Step 1: Understand the Geometry
We have two circles with centers A and B that intersect at points C and D. The center B of the smaller circle lies on the circumference of the larger circle centered at A.

### Step 2: Identify Tangents
Since B lies on the circumference of the larger circle, the lines CB and BD are tangents to the larger circle at points C and D respectively. Therefore, angles CBM and BDN are right angles (90 degrees).

### Step 3: Relate Angles
We know that angle CMD is given as x degrees. The angle subtended at the center A by the chord CD is twice the angle subtended at any point on the circumference (which is angle CMD). Therefore, we can write:
\[ \angle CAB = 2x \]

### Step 4: Use the Sum of Angles in Quadrilateral
In quadrilateral ACBD, the sum of the angles is equal to 360 degrees. The angles we have are:
- Angle CAB = 2x
- Angle CBM = 90 degrees
- Angle BDN = 90 degrees
- Angle CND (which we will express in terms of angle CAD)

Thus, we can write the equation:
\[ 90 + 90 + 2x + \angle CND = 360 \]

### Step 5: Solve for Angle CND
From the equation above, we can simplify:
\[ 180 + 2x + \angle CND = 360 \]
Subtracting 180 and 2x from both sides gives:
\[ \angle CND = 180 - 2x \]

### Step 6: Relate Angle CND to Angle CAD
The angle CND is half of angle CAD (since it subtends the same arc CD). Therefore, we can write:
\[ \angle CND = \frac{1}{2} \angle CAD \]
Substituting the expression we found for angle CND:
\[ 180 - 2x = \frac{1}{2} \angle CAD \]

### Step 7: Solve for Angle CAD
To find angle CAD, we multiply both sides by 2:
\[ 2(180 - 2x) = \angle CAD \]
This simplifies to:
\[ \angle CAD = 360 - 4x \]

### Step 8: Find Angle DAC
Since angle DAC is the same as angle CAD, we can conclude that:
\[ \angle DAC = 360 - 4x \]

### Final Answer
Thus, the measure of angle DAC in terms of x is:
\[ \angle DAC = 360 - 4x \]

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