On a circle of radius 7 units, PQ and QR are chords of length 7 units each. What is the length of the chord PR in units?

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Given a circle with radius 7 units, chords PQ and QR each have length 7 units. We need to find the length of chord PR.

Step 1: Since PQ and QR are chords of equal length (7 units), triangle PQR is isosceles with PQ = QR = 7 units.

Step 2: The radius of the circle is 7 units. The length of each chord PQ and QR is equal to the radius.

Step 3: The perpendicular from the center O to a chord bisects the chord. Let M and N be midpoints of PQ and QR respectively.

Step 4: Since PQ = 7, PM = MQ = 3.5 units.    
Similarly, RN = NQ = 3.5 units.

Step 5: The distance from the center O to chord PQ is given by:  
Using right triangle OMP,  
OM = sqrt(OP^2 - PM^2) = sqrt(7^2 - 3.5^2) = sqrt(49 - 12.25) = sqrt(36.75) = 6.062 units approximately.

Similarly, ON = 6.062 units.

Step 6: Since chords PQ and QR are equal and subtend equal angles at the center, angle POQ = angle QOR.

Step 7: The angle subtended by chord PQ at the center is given by:  
Using the formula for chord length c = 2r sin(θ/2), where θ is the angle subtended at the center,
7 = 2 * 7 * sin(θ/2) => sin(θ/2) = 7 / 14 = 0.5 => θ/2 = 30° => θ = 60°.

Step 8: Therefore, angle POQ = angle QOR = 60°. Hence, angle POR = 120°.

Step 9: Triangle PQR is inscribed in the circle, and chord PR subtends angle POR at the center.

Step 10: Length of chord PR = 2r sin(angle POR / 2) = 2 * 7 * sin(60°) = 14 * (√3 / 2) = 7√3.

Step 11: The question options are in the form of fractions with √3 in the denominator. Convert 7√3 to this form:
7√3 = (7 * 3) / √3 = 21 / √3.

Hence, the length of chord PR is 21 / √3 units.

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