Find an acute angle between the tangent and the chord drawn through the tangency point

Find an acute angle between the tangent and the chord drawn through the tangency point if the chord divides the circle in a ratio of 2: 7.

Let AB be a given chord. The tangent passes through point A, O is the center of the circle.

Let’s find the degree measure of each arc. Let the arcs be 2x and 7x, then:

2x + 7x = 360 °;

9x = 360;

x = 360/9 = 40 °.

Then the arcs are 2x = 2 * 40 ° = 80 ° and 7x = 7 * 40 ° = 280 °.

Consider a triangle AOB: the central angle AOB is equal to the arc value, it is 80 °. ОА = ОВ (these are the radii). Hence, the triangle AOB is isosceles, the angle BAO = angle ABO = (180 ° – 80 °): 2 = 50 °.

The angle between tangent and radius is 90 °, so the angle between tangent and chord is 90 ° – 50 ° = 40 °.

Answer: The angle between the chord and the tangent is 40 °.