Through point M, tangents MK and ME are drawn to a circle centered at point O, where K and E

| February 14, 2021 | education

Through point M, tangents MK and ME are drawn to a circle centered at point O, where K and E are tangency points, angle OMC = 30 degrees, angle MK = 6 cm. Find the length of the chord KE.

Radii OE and OK, by the property of tangents, form right angles with tangents, then triangles OKM and OEM are rectangular.

In the right-angled triangle OKM, we define the leg OK, which is the radius of the circle.

tg30 = OK / MK.

OK = MK * tg30 = 6 * √3 / 3 = 2 * √3 cm.

The OM segment is the bisector of the KME angle, then the KME angle = 2 * 30 = 600.

In the OKME quadrangle, the КОЕ angle is: КОЕ = (360 – 90 – 90 – 60) = 1200.

The OKE triangle is isosceles, OK = OE = 2 * √3 cm, then by the cosine theorem:

KE ^ 2 = OK ^ 2 + OE ^ 2 – 2 * OK * OE * Cos120 = 12 + 12 – 2 * 2 * √3 * 2 * √3 * (-1/2) = 24 + 12 = 36.

KE = 6 cm.

Answer: The length of the KE chord is 6 cm.



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