# 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 °.