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In the diagram below, chords $\overline{AB}$ and $\overline{CD}$ are perpendicular, and meet at $X.$  Find the diameter of the circle.

 Dec 17, 2023
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To find the diameter of the circle, we can leverage the properties of perpendicular chords and right triangles within the circle. Here's how:

 

Locate the circle's center:

 

Since lines AB and CD are perpendicular chords, their intersection point X lies at the center of the circle.

 

Form right triangles:

 

Triangle AXB and triangle CXD are right triangles with X as the right angle.

 

Apply the Pythagorean theorem:

 

In triangle AXB, we know AB = 5 and since X is the center, AX = XB = radius of the circle (let's call it r). Applying the Pythagorean theorem:

 

r^2 = (5/2)^2 --> r^2 = 25/4 --> r = sqrt(25/4) = 5/2

 

Similarly, in triangle CXD, we have CD = 3 and CX = DX = r. Applying the Pythagorean theorem:

 

r^2 = (3/2)^2 --> r^2 = 9/4 --> r = sqrt(9/4) = 3/2

 Dec 17, 2023

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