Question

select the correct answer from each drop-down menu.
three students are trying to find the diameter of circle c.

circle diagram with points x, y, w, z, and center c; segments wz=27, xz=36, yz=36, right angle at z

gregory believes the diameter is equal to the length of chord xy. maria believes chord wy can be added to create the right triangle wzy. she also thinks the hypotenuse of δwzy has a length equal to the radius of the circle.
jordan believes segment wz lies on the diameter of the circle, and that if the diameter is drawn, he can make an equation relating the pieces of that diameter of the circle to the pieces of \\(\overline{xy}\\).

the student with the correct approach to finding the diameter is drop-down: gregory, maria, jordan. the diameter of the circle is drop-down units.

Explanation:

Step1: Identify correct student

Jordan uses the Power of a Point theorem (for a point inside a circle: product of segments of one chord equals product of segments of another chord). Let diameter be \(d\), radius \(r = \frac{d}{2}\). Let \(CZ = x\), so \(ZW = r - x\) or \(x - r\)? Wait, Power of Point Z: \(ZX \times ZY = ZW \times Z(\text{other segment on diameter})\). \(ZX=36\), \(ZY=36\), \(ZW=27\), let the diameter pass through Z, so segments are \(ZW=27\) and \(d - 27\)? No: Power of Point Z inside circle: \(ZX \cdot ZY = ZW \cdot ZV\) where \(V\) is opposite end of diameter through Z. Let \(ZV = k\), so \(36 \times 36 = 27 \times k\) → \(k = \frac{36^2}{27} = 48\). Then diameter is \(ZW + ZV = 27 + 48 = 63\). So Jordan is correct.

Step2: Calculate diameter

Using Power of Point Z: \(ZX \times ZY = ZW \times ZV\) → \(36×36=27×ZV\) → \(ZV=48\). Diameter = \(27+48=63\).

Answer:

Jordan; 63