A chord AB = 6 cm is drawn in a circle of radius 5 cm. A point P is marked on the line AB outside the chord
A chord AB = 6 cm is drawn in a circle of radius 5 cm. A point P is marked on the line AB outside the chord so that AP: PB = 5: 2. Find the distance from point P to the center of the circle.
Let the length of the segment AP = 5 * X cm, then, by condition, PB = 2 * X cm.
The length of the segment AB = AP – BP.
6 = 5 * X – 2 * X.
3 * X = 6 cm.
X = 6/3 = 2 cm.
PB = 2 * 4 = 4 cm, then AR = 6 + 4 = 10 cm.
Segments ОА = ОВ = R = 5 cm. In the isosceles triangle AOB, we draw the height ОН, which is also its median, then АН = ВН = 6/2 = 3 cm.
Then OH ^ 2 = OB ^ 2 – BH ^ 2 = 25 – 9 = 16.
OH = 4 cm.
PH = AP – AH = 10 – 3 = 7 cm.
The triangle OHP is rectangular then, by the Pythagorean theorem, OP ^ 2 = PH ^ 2 + OH ^ 2 = 49 + 16 = 65.
RR = √65 cm.
Answer: From point P to the center of the circle √65 cm.