Questions: In the diagram at right, A is the center of the circle and CD ⊥ AB. We are given that angle ACD = 2 angle DCB. (a) Let angle DCB = x. Find expressions for angle ABC and angle BAC in terms of x. (b) Find angle ABC.

In the diagram at right, A is the center of the circle and CD ⊥ AB. We are given that angle ACD = 2 angle DCB.
(a) Let angle DCB = x. Find expressions for angle ABC and angle BAC in terms of x.
(b) Find angle ABC.

Transcript text: 3.40 In the diagram at right, $A$ is the center of the circle and $\overline{\mathrm{CD}} \perp \overline{\mathrm{AB}}$. We are given that $\angle A C D=2 \angle D C B$. (a) Let $\angle D C B=x$. Find expressions for $\angle A B C$ and $\angle B A C$ in terms of $x$. (b) Find $\angle A B C$.

Solution

Solution Steps

Step 1: Understanding the Given Information

$ A $ is the center of the circle.
$ CD \perp AB $.
$ \angle ACD = 2 \angle DCB $.

Step 2: Define the Given Angle

Let $ \angle DCB = x $.

Step 3: Express $ \angle ACD $ in Terms of $ x $

Given $ \angle ACD = 2 \angle DCB $, we have: \[ \angle ACD = 2x \]

Step 4: Find $ \angle BAC $ in Terms of $ x $

Since $ A $ is the center of the circle, $ \angle BAC $ is the central angle subtended by the arc $ BC $.
The central angle $ \angle BAC $ is twice the inscribed angle $ \angle BDC $ (which is $ x $): \[ \angle BAC = 2x \]

Step 5: Find $ \angle ABC $

In the right triangle $ \triangle ABC $, $ \angle ABC $ is complementary to $ \angle BAC $ because $ \angle ACB = 90^\circ $: \[ \angle ABC = 90^\circ - \angle BAC \]
Substituting $ \angle BAC = 2x $: \[ \angle ABC = 90^\circ - 2x \]

Final Answer

$ \angle ABC = 90^\circ - 2x $
$ \angle BAC = 2x $

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