Given:
• A(6,6), B(0,4), C(4,0)
• O is the origin (0,0)
• Segment AO bisects angle ∠BOC
a
Find angle ∠AOC
Note:
• B is on the positive Y-axis (since x=0)
• C is on the positive X-axis (since y=0)
• What is the angle between the X-axis and the Y-axis?
∠AOC = degrees
Correct! 90° / 2 = 45°
Explanation:
• OB is the positive Y-axis, OC is the positive X-axis
• The angle between the axes: ∠BOC = 90°
• AO bisects angle ∠BOC
• Therefore ∠AOC = 90° / 2 = 45°
b
Prove that triangles ΔABO and ΔACO are congruent
Hint: Think about the lengths of segments OB and OC
• OB = distance from the origin to B(0,4) = ?
• OC = distance from the origin to C(4,0) = ?
What is the length of OB?
OB =
What is the length of OC?
OC =
Choose the congruence theorem:
Excellent! The triangles are congruent by SAS
Explanation:
• OB = 4 (distance from (0,0) to (0,4))
• OC = 4 (distance from (0,0) to (4,0))
• Therefore OB = OC
• AO is a common side
• ∠BOA = ∠AOC = 45° (since AO bisects ∠BOC)
• Therefore ΔABO ≅ ΔACO by SAS (Side-Angle-Side)
Well done! You completed Question 43!
You understood analytic geometry and triangle congruence proofs!