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

kittykat Dec 17, 2023

#1**0 **

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

BuiIderBoi Dec 17, 2023