A chord equal to 0.5 r is drawn in a circle of radius r. A tangent to the circle is drawn through one end of the chord

A chord equal to 0.5 r is drawn in a circle of radius r. A tangent to the circle is drawn through one end of the chord, and a secant, parallel to the tangent, through the other. Find the distance between the tangent and the secant.

Let’s draw the radii ОА and ОВ. The radius OA is perpendicular to the tangent AC, and since, by condition, AC is parallel to BD, OA is perpendicular to BD.

Triangle AOB is isosceles, AO = BO = r, AB = r / 2. The segment BH is the height of triangle AOB, then triangles ABH and OBН are rectangular. Let the length AH = X cm, then OH = (AO – X) = (r – X).

From the right-angled triangles ABH and OBН, we express the height BH.

BH ^ 2 = AB ^ 2 – AH ^ 2 = (r / 2) ^ 2 – (X) ^ 2.

BH ^ 2 = OB ^ 2 – OH ^ 2 = r ^ 2 – (r – X) ^ 2.

Then: (r / 2) ^ 2 – (X) ^ 2 = r ^ 2 – (r – X) ^ 2.

r ^ 2/4 – X ^ 2 = r ^ 2 – r ^ 2 + 2 * r * X – X ^ 2.

r ^ 2/4 = 2 * r * X.

2 * X = r / 4.

X = AH = r / 8.

Answer: The distance between the tangent and the secant is r / 8.