Two congruent circles Γ1 and Γ2 each have radius 213, and the center of Γ1 lies on Γ2. Suppose Γ1 and Γ2 intersect at A and B. The line through A perpendicular to AB meets Γ1 and Γ2 again at C and D, respectively. Find the length of CD.

426 213(root(3))

Actually It would be 426(root(3))

it would be the root of 837

To find the length of CD, we need to use the properties of the intersecting circles and the given information about the line through A.

Step 1: Draw a diagram
Start by drawing the two intersecting circles with the given radius of 213. Label the centers of the circles as O₁ and O₂, and the points of intersection as A and B.

Step 2: Identify the perpendicular line
Draw a line through point A that is perpendicular to the line segment AB. Label the points of intersection of this line with the circles as C and D.

Step 3: Find the length of AD
Since AD is a radius of circle Γ₂, its length is also 213.

Step 4: Find the length of AC
To find the length of AC, we can use the properties of the perpendicular line. The line through A perpendicular to AB is the same as the line passing through O₁ and O₂, since the centers of the circles lie on this line.

Since the circles are congruent, the distance from O₁ to O₂ is twice the radius or 2 * 213 = 426. Therefore, AC is also equal to 426.

Step 5: Calculate the length of CD
The length of CD can be obtained by subtracting the length of AC from the length of AD.

CD = AD - AC
= 213 - 426
= -213

Therefore, the length of CD is -213 units.

Note: The negative sign indicates that CD lies in the opposite direction from A.