The angle between two tangents from a point to a circle is 82 degree.What is the length of one of these tangents if the radius of the circle is 80mm?

radius perpendicular to tangent to circle so

tan 41 = 80/x

It was helpful to me

To find the length of one of the tangents, we can use the relationship between the angle and the tangent lines to the circle.

Step 1: Recall that the tangents drawn from an external point to a circle are equal in length.

Step 2: Let AB and AC be the two tangents drawn from point A to the circle.

Step 3: Draw radius OA from the center of the circle to point A.

Step 4: The angle between the two tangents, which is given as 82 degrees, is equal to the angle OAB (angle between the radius and one of the tangents).

Step 5: In a circle, the angle between a radius and a tangent line is 90 degrees.

Step 6: Thus, angle OAB is 90 - 82 = 8 degrees.

Step 7: Now, we have a right triangle OAB with angle OAB as 8 degrees and the hypotenuse OA as the radius of the circle, which is given as 80 mm.

Step 8: To find the length of the tangent AB, we can use trigonometry. We know that tangent(Angle) = Opposite/Adjacent.

Step 9: In this triangle, the adjacent side is AB (the length of the tangent we want to find), and the opposite side is OA (the radius of the circle).

Step 10: We can rearrange the formula to solve for AB: AB = OA * tan(Angle).

Step 11: Substituting the given values: AB = 80 * tan(8).

Step 12: Using a calculator, we find: AB ≈ 11.032 mm.

Therefore, the length of one of the tangents is approximately 11.032 mm when the radius of the circle is 80 mm.

To find the length of one of the tangents, we can use the trigonometric relationship between the angle and the length of the tangent.

Let's denote the length of one of the tangents as 'x'.

Using the given angle of 82 degrees, we can use the tangent function to determine the relationship between the length of the tangent and the angle:

tan(82°) = x / 80

To solve for x, we can rearrange the equation:

x = tan(82°) * 80

Now, let's calculate the length of the tangent:

Using a scientific calculator, we find that tan(82°) ≈ 7.11537

So, substituting this value into the equation for x:

x ≈ 7.11537 * 80 ≈ 569.23096

Therefore, the length of one of the tangents is approximately 569.23 mm.