Chords AB and CD are drawn in a circle with center O. Lines AB and CD are perpendicular

| July 20, 2021 | education

Chords AB and CD are drawn in a circle with center O. Lines AB and CD are perpendicular and intersect at a point M that lies outside the circle. with AM = 36, BM = 6, CD = 4√46. find OM.

Let us draw from the center O perpendiculars OK and OH to the chords.

The segment OK divides the chord CD in half, then CK = CD / 2 = 4 * √46 / 2 = 2 * √46 cm.

OH divides the chord AB in half, then AH = BH = AB / 2 = (36 – 6) / 2 = 15 cm

Quadrangle MKOН is a rectangle, then OK = MН = ВН + ВM = 15 + 6 = 21 cm.

From the right-angled triangle РНС, we determine the length of the hypotenuse of the OС.

OС ^ 2 = OK ^ 2 + СK ^ 2 = 441 + 184 = 625.

OС = 25 cm.

And the right-angled triangle AOН will determine the length of the leg OH.

OH ^ 2 = OA ^ 2 – AH ^ 2 = 625 – 225 = 400.

OH = 20 cm.

From the right-angled triangle ОНМ we determine the length of the hypotenuse ОМ.

OM ^ 2 = MH ^ 2 + OH ^ 2 = 441 + 400 = 841.

OM = 29 cm.

Answer: The length of the segment OM is 29 cm.



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