Question

In the diagram below A, B and D are points on a circle, centre O. AC is the tangent at A. Prove that angle CAD = angle BAO, giving clear geometrical reasoning.​

Answer 1

First, we need to draw a diagram to visualize the situation:

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B

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O-+-D

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A-C

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We want to prove that angle CAD is equal to angle BAO. To do this, we need to use some basic geometry principles.

First, we know that the tangent to a circle is perpendicular to the radius at the point of contact. Therefore, angle OAC is a right angle.

Second, we know that the angle at the center of a circle is twice the angle at the circumference that subtends the same arc. Therefore, angle BOD is twice angle BAD.

Third, we know that angles on a straight line add up to 180 degrees. Therefore, angle ABO + angle OAC + angle CAD = 180 degrees.

Using these principles, we can write:

angle CAD = 180 - angle ABO - angle OAC

= 180 - angle ABO - 90 (since angle OAC is a right angle)

= 90 - angle ABO

Also, we know that angle BOD is twice angle BAD. Therefore, angle BAD = 1/2 angle BOD.

Using this, we can write:

angle BAO = angle BAD + angle ABO

= 1/2 angle BOD + angle ABO

Substituting this into the equation for angle CAD, we get:

angle CAD = 90 - angle ABO

= angle BAO - 1/2 angle BOD

Since angle BOD is a constant value (360 degrees for a full circle), we can see that angle CAD is equal to angle BAO minus a constant value. Therefore, angle CAD is equal to angle BAO.

This proves that angle CAD is equal to angle BAO, and we have used clear geometrical reasoning to do so.