Question

Given AB,CD are diameters. Prove: BD=CA
A. BD=AC
B. BD>CA
C. BDD. CA=BD

Answer 1

In this question, we are given diameters AB and CD. We need to prove several statements based on these diameters, such as BD = CAA, BD = ACB, BD > CAC, and BD = CA. However, only the first two statements are true.

Step-by-step explanation:

In this question, we are given AB and CD as diameters. We need to prove several statements based on these diameters. Let's go through each step:

1. BD = CAA: Since AB is a diameter, angle BAD is a right angle. Therefore, angle BDA is also a right angle. In triangle CAA, angle CAA is a right angle because AC is a diameter. Since angle BDA and angle CAA are both right angles, BD and AC are parallel. This implies that BD = CAA.
2. BD = ACB: In triangle CAA, angle CAA is a right angle. In triangle CAB, angle CAB is also a right angle because AB is a diameter. Therefore, triangle CAA ~ triangle CAB by AA similarity. This implies that angle CBA = angle CAA. Since triangle CBD has angle BDA = 90 degrees and angle CAA = angle CBA, triangle CBD and triangle CAA are similar by AA similarity. Therefore, angle CBD = angle DAA. Since angle CBD + angle DAA = 180 degrees, angle CBD = angle DAA = 90 degrees. This implies that triangle CBD is a right triangle. In a right triangle, the hypotenuse is the longest side, so BD > CA. Therefore, BD = CAA = ACB.
3. BD > CAC: This statement is not true. From the previous step, we know that BD = CAA = ACB. Therefore, BD is equal to CAC, not greater than it.
4. BD = CA: This statement is not true. From the previous step, we know that BD = CAA = ACB. Therefore, BD is equal to ACB, not CA.