I am ask to write a two column proof. Given that AD is the diameter of circle O, and DC is tangent to O at D. Prove that Triangle ABD ~ (congruent) to triangle ADC
pretty hard to prove, knowing nothing about point B,..
To prove that Triangle ABD is similar to Triangle ADC, we need to show that their corresponding angles are congruent.
Here is a two-column proof that you can use to demonstrate this:
Statement | Reason
-------------------------------------------------------
1. AD is the diameter of circle O | Given
2. DC is tangent to O at D | Given
3. Angle ABD is a right angle | Definition of a diameter
4. Angle ADC is a right angle | Tangent and radius are perpendicular
5. Angle ABD is congruent to Angle ADC | Both are right angles
6. Triangle ABD is similar to Triangle ADC | Definition of similar triangles
Explanation of each statement:
1. The given statement tells us that AD is the diameter of circle O. This means that AD passes through the center of the circle and divides it into equal halves.
2. The given statement tells us that DC is tangent to O at D. This means that DC touches the circumference of the circle at D and is perpendicular to the radius drawn to D.
3. The definition of a diameter states that it passes through the center of the circle and forms a right angle with any tangent drawn to the circle at its endpoints. Therefore, Angle ABD is a right angle.
4. The tangent drawn to a circle is perpendicular to the radius drawn to the point of tangency. This means that Angle ADC is also a right angle.
5. Both Angle ABD and Angle ADC are right angles, and by definition, right angles are congruent.
6. When two triangles have corresponding angles that are congruent, we can conclude that the triangles are similar. Therefore, Triangle ABD is similar to Triangle ADC.
By following the logical steps in this two-column proof, you have demonstrated that Triangle ABD is similar to Triangle ADC based on the given conditions.