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Geometry

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I need help plz

Let AB be a diameter of a circle, and let C be a point on the circle such that AC=8 and BC=4 The angle bisector of ACB intersects the circle at point M Find CM.

Mar 7, 2023

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To solve this problem, we can use the angle bisector theorem and the properties of cyclic quadrilaterals.

First, let's draw a diagram:B / \ / \ / \ / \ / \ / \ A-------------C

Since AB is a diameter of the circle, angle ACB is a right angle (90 degrees).

Let AM be the angle bisector of angle ACB. Then, by the angle bisector theorem, we have:

AC AM -- = ----- BC BM

Substituting the given values, we get:

8 AM -- = --- 4 BM

Simplifying, we get:

AM = 2BM

Now, let's draw the line CM and label the point where it intersects the circle as D:B / \ / \ / D \ / \ / \ / \ A-------------C | | M

Since AM is twice BM, we can let BM = x and AM = 2x. Then, using the Pythagorean theorem in triangle ABC, we have:

AC^2 + BC^2 = AB^2 8^2 + 4^2 = AB^2 AB = 2sqrt(5)

Since AD and BD are radii of the circle, we have AD = BD = AB/2 = sqrt(5).

Now, let's use the properties of cyclic quadrilaterals to find the length of CD. Since angle CMB is an inscribed angle that intercepts arc CD, we have:

angle CMB = 1/2 * arc CD

Also, angle CAM and CBM are equal because they are opposite angles of an isosceles triangle. Therefore, angle AMC = 180 - 2CAM = 180 - 2CBM. Since angle AMC is an inscribed angle that intercepts arc CD, we have:

angle AMC = 1/2 * arc CD

Adding these two equations, we get:angle CMB + angle AMC = arc CD 1/2 * arc CD + 1/2 * arc CD = arc CD arc CD = angle ACB = 90 degrees

Therefore, CD is a diameter of the circle, and CM is the radius. Since CM = CD/2, we have:CM = AB/2 - AC/2 = (2sqrt(5))/2 - 8/2 = sqrt(5) - 4

Therefore, CM = sqrt(5) - 4. My AARP Medicare

Mar 7, 2023
edited by Amandahertzog  Mar 7, 2023