If two tangents inclined at an angle 60 degree are drawn to a circle of radius 6 CM then find the length of each tangent?

Question

Answer

Answer

Let the tangent AB and AC are two tangents drawn from external point A and a circle with centre O.
join AO.
Now in triangle ABO
tan30=OB/BA
1/√3=6/AB
AB=6√3cm

Hence length of tangents is 6√3cm

Or u can rationalized it .

That's it☺

Answer

Answer

The length of each tangent is 6√3

Answer

Answer

Problem: If two tangents inclined at an angle of 60 degrees are drawn to a circle of radius 6 cm, find the length of each tangent.

Solution:

To find the length of each tangent, we can use the properties of tangents and the given angle.

Properties of Tangents:
1. A tangent is perpendicular to the radius at the point of contact.
2. The radius drawn to the point of tangency bisects the tangent.

Let's break down the problem into smaller steps to find the length of each tangent.

Step 1: Draw a diagram:

We start by drawing a circle with a radius of 6 cm. Then, we draw two tangents inclined at an angle of 60 degrees. Label the points of contact as A and B.

Step 2: Identify the right triangles:

Since the tangents are perpendicular to the radius at the points of contact, we can identify two right triangles formed by the radii and the tangents. Let's label the right triangles as ΔOAP and ΔOBP.

Step 3: Identify the common angle:

In both triangles, the angle between the radius and the tangent is 90 degrees. This angle is common to both triangles.

Step 4: Measure the angle:

Given that the angle between the tangents is 60 degrees, we can conclude that the angle between the radii is also 60 degrees. This is because the sum of angles around a point is 360 degrees, and the angle between the tangents and the radii is supplementary.

Step 5: Apply trigonometry:

In ΔOAP, we can apply trigonometry to find the length of the tangent OA. We have the angle (60 degrees) and the adjacent side (radius = 6 cm). We can use the cosine function:

cos(60) = adjacent/hypotenuse
cos(60) = 6/OA

Simplifying the equation, we get:
OA = 6/cos(60)

Similarly, in ΔOBP, we can find the length of the tangent OB using the same approach.

Step 6: Calculate the values:

Using a scientific calculator, we can calculate the value of cos(60) as 0.5. Substituting this value in the equation, we find:

OA = 6/0.5 = 12 cm
OB = 6/0.5 = 12 cm

Hence, the length of each tangent is 12 cm.

Summary:

Given a circle with a radius of 6 cm and two tangents inclined at an angle of 60 degrees, we used the properties of tangents and trigonometry to find the length of each tangent. By identifying the right triangles formed by the radii and the tangents, we determined the common angle between them. Applying trigonometry, we calculated the length of each tangent to be 12 cm.

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