draw the tangents to its end

Question

Draw a chord of length 2.3 cm in a circle of radius 3.1 cm and draw the tangents to its both ends

Answer 1

Here is how to draw a fairly accurate sketch.

If you draw radii to the endpoints of the chord of length 2.3, you create an isosceles triangle with equal sides 3.1 and a base of 2.3. There are a couple ways to determine the central angle created by those two radii, both involving trigonometry.

You could use the Law of Cosines:

2.3² = 3.1² + 3.1² - 2(3.1)(3.1)cos(θ)

Solving for θ ≈ 43.55°.

Or draw an altitude from the center of the circle to the midpoint of the chord creating a right triangle with one leg 2.3/2 = 1.15 and a hypotenuse of 3.1. So, sin(θ) = 1.15/3.1, θ = 2sin^-1(1.15/3.1) ≈ 43.55°. You have to multiply by two because the acute angle created by a radius and the altitude is 1/2 θ.

Draw a circle with radius 3.1 cm, draw two radii with a central angle 43.55°, connect the endpoints of those radii to generate a chord of length 2.3,

The tangent lines at those endpoints are perpendicular to the radii at the points of tangency.

This graph on Desmos shows the result:

desmos.com/calculator/soxgkyz8oz