SSC CGL 2024 PYQ — Question 75

The correct answer is option 4 i.e. $21\over\sqrt3$.

To find the length of the chord PR on a circle of radius 7 units, where PQ and QR are chords of length 7 units each, we can use geometric properties and trigonometry. Let's proceed step by step.

Identify the Geometry:

The circle has a radius r = 7 units.

Chords PQ and QR each have a length of 7 units.

We need to find the length of chord PR.

Determine the Central Angles:

Since PQ and QR are both equal to the radius of the circle, triangles OPQ and OQR are equilateral triangles, where O is the center of the circle.

Therefore, the central angles ∠POQ and ∠QOR are both 60°.

Calculate the Angle ∠POR:

⇒ ∠POR = 60° + 60° = 120°

The angle ∠POR is the sum of ∠POQ and ∠QOR.

Use the Law of Cosines in Triangle POR:

⇒ PR$^2$ = PO$^2$ + OR$^2$ − 2⋅PO⋅OR⋅cos(∠POR)

Substituting the known values:

⇒ PR$^2$= 7$^2$ + 7$^2$ − 2⋅7⋅7⋅cos(120°)

Since cos(120∘)=−21​:

⇒ PR$^2$ = 49 + 49 − 2⋅7⋅7⋅(−$\frac{1}{2}$​)

Simplify the expression:

⇒ PR$^2$ = 49 + 49 + 49 = 147

Taking the square root of both sides:

⇒ PR =$\sqrt{147}$ = 7$\sqrt3$​

⇒ In triangle POR, we know the lengths of sides PO and OR (both equal to the radius 7 units) and the angle ∠POR=120°.

We can use the Law of Cosines to find the length of side PR.

Express the Answer in the Given Format:

⇒ The length of chord PR is 7$\sqrt3$​, which can be written as $\frac{21}{\sqrt3}$​ after rationalizing the denominator.

Therefore, the correct answer is $\frac{21}{\sqrt3}$