Question

AB is the diameter of the circle with centre O. CB is the tangent to the circle at B. AC intersect the circle at C. If the radius of the circle is 6cm and AG=8 then the length of BC

Answer 1

If the radius of the circle is 6 cm and AG = 8, then the length of BC is 13.4 cm.

Step-by-step explanation:

Referring to the figure attached below

The radius of the circle, AO = 6 cm

∴ The diameter of the circle, AB = 6*2 = 12 cm

AG = 8 cm

Join points B and G.

Step 1:

We know that the angle subtended by the diameter of the circle on any point of the circle is 90°.

∴ ∠AGB = 90°

Let’s assume ∠GAB = “θ”.

Applying the trigonometry properties of triangles in ∆AGB, we get

cos θ = $\frac{Base}{Hypotenuse} = \frac{AG}{AB}$

⇒ cos θ = $\frac{8}{12}$

⇒ cos θ = $\frac{2}{3}$

⇒ θ = cos⁻¹ (0.667)

⇒ θ = 48.16°

Step 2:

We also know that the angle made by the tangent and the radius of the circle is equal to 90°.

∴ ∠BCO = ∠BCA = 90°

Now, consider ∆ABC and apply the trigonometry properties of triangles, we get

tan θ = $\frac{Perpendicular}{Base} = \frac{BC}{AB}$

⇒ tan 48.16° = $\frac{BC}{12}$ …… [here ∠GAB = ∠CAB = θ = 48.16°]

⇒ BC = 1.116 * 12

⇒ BC = 13.39 cm ≈ 13.4 cm

Thus, the length of BC is 13.4 cm .

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