State with reasons that angle ACB is bisected by line OC

In summary, the problem involves finding the relation between triangles XYC and XZC, where X is a point outside of circle C and Y and Z are the points of tangency on the circle. After correcting some errors in the construction, it is determined that angle ACB is bisected by line OC due to the congruence of triangles XYC and XZC. This is because of the radius of the circle and the common line CD, creating a 90 degree angle at the tangent line.
  • #1
mathlearn
331
0
Here is the Problem

View attachment 6180

Here is the construction, Hoping that I have done it correct "State with reasons that $\angle$ ACB is bisected by line OC"

View attachment 6181

Many THanks (Party)
 

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  • #2
AB is given as 6 cm and the circle should be tangent to AB.
 
  • #3
greg1313 said:
AB is given as 6 cm and the circle should be tangent to AB.

Yes I updated the diagram correcting that error :) As AB = AC as they are tangents drawn to the circle from external point A. $\therefore \angle{ABC}=\angle{ACB}$

But still how can it be proved that
State with reasons that $\angle {ACB}$ is bisected by line OC"
 
  • #4
Let $X$ be any point outside of a circle $C$. Let the points $Y$, $Z$ be the points of tangency at the circle $C$ and the lines $XY$ and $XZ$. What can be said about triangles $XYC$ and $XZC$ ?
 
  • #5
mathlearn said:
Here is the Problem
Here is the construction, Hoping that I have done it correct "State with reasons that $\angle$ ACB is bisected by line OC"
Many THanks (Party)

greg1313 said:
Let $X$ be any point outside of a circle $C$. Let the points $Y$, $Z$ be the points of tangency at the circle $C$ and the lines $XY$ and $XZ$. What can be said about triangles $XYC$ and $XZC$ ?

Thanks but I doubt whether the Line $OC$ bisects angle $ACB$ which is really inside the triangle ? (Thinking) Have I done the construction wrong? Is the type of the circle correct?
 
  • #6
See post #2.

The relation that is to be found is true in general, which is what I was trying to get at in post #4. Make the necessary corrections and apply the hints I gave to state your answer (with reasons).
 
  • #7
greg1313 said:
See post #2.

The relation that is to be found is true in general, which is what I was trying to get at in post #4. Make the necessary corrections and apply the hints I gave to state your answer (with reasons).
View attachment 6185

Many THanks :)
 

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  • #8

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  • #9
:D Thank you very much , I was struggling a long here for around a day or two

The sides was produced in the other direction

And Greg I think the reason for the bisection is the congruence of those two triangles due to the radius of the circle at the tangent and the common line CD and the 90 degree angle formed at the tangent line and the radius

Many THanks (Smile)
 
  • #10
That's correct. Good work! :)
 

Related to State with reasons that angle ACB is bisected by line OC

1. What does it mean for an angle to be bisected?

When a line or ray divides an angle into two equal parts, it is called the angle bisector. This means that the angle is divided into two congruent angles, or angles with the same measure.

2. How can I prove that angle ACB is bisected by line OC?

To prove that angle ACB is bisected by line OC, you can use the angle bisector theorem or the definition of an angle bisector. The angle bisector theorem states that if a line divides an angle into two congruent angles, then it divides the opposite side into two segments that are proportional to the adjacent sides of the angle. The definition of an angle bisector simply states that the line divides the angle into two equal parts.

3. What is the importance of an angle bisector in geometry?

The angle bisector is an important tool in geometry as it helps in solving problems involving angles and triangles. It also helps in constructing various geometric figures, such as perpendicular lines and parallel lines. Moreover, the angle bisector theorem has many real-world applications, such as in navigation, architecture, and engineering.

4. Can an angle be bisected by more than one line?

Yes, an angle can be bisected by multiple lines. As long as the lines divide the angle into two equal parts, they are considered angle bisectors. This is similar to how a line can have multiple perpendicular bisectors, as long as they intersect at the same point on the line.

5. Is it possible for an angle to have more than one angle bisector?

No, an angle can only have one angle bisector. This is because the angle bisector divides the angle into two equal parts, and if there were more than one angle bisector, the angle would be divided into more than two parts, which is not possible. However, an angle can have multiple bisectors, such as a perpendicular bisector or a median, as long as they are different types of bisectors and not angle bisectors.

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