Two mutually perpendicular chords are drawn from one point of the circle.
Two mutually perpendicular chords are drawn from one point of the circle. Find their lengths if they are 2 cm and 5 cm from the center of the circle.
Let AB and AC be two chords, AB is perpendicular to AC. Let us draw from the center of the circle O the perpendicular OH to the chord AB and the perpendicular OE to the chord AC. OE and OH will be the distance from the center of the circle to the chords. Let OH = 5 cm, and OE = 2 cm.
Consider a quadrilateral AEOH: AH is perpendicular to AE (since AB is perpendicular to AC), OH is perpendicular to AB and OE is perpendicular to AC, so AEOH is a rectangle.
Therefore, OH = AE = 5 cm, OE = AH = 2 cm.
Consider the triangle AOB: OA = OB (radii), triangle AOB is isosceles, which means that OH is not only the height, but also the median:
AB = AH * 2 = 2 * 2 = 4 (cm).
Similarly, AC = AE * 2 = 5 * 2 = 10 (cm).
Answer: the lengths of the chords are 4 cm and 10 cm.