Question 690163: This problem is the first three dimensional problem I have seen, and I'm having a lot of trouble with it. I would have tried it, but I didn't know where to start! The problem is this:
Given:
segment AP is perpendicular to plane M
segment AB is congruent to segment AC
Prove:
triangle BPC is isosceles.
Diagram:
(my best description)
A plane M is shwn.
B, P, and C are all points on plane M. They connected to form a triangle.
A is a point that looks to be straight above point P. From the given information, I can deduce that point A actually is right above point P.
Together, points A, B, C, and P create a triangular prism.
What I've reasoned:
angle ABC is congruent to angle ACB because of the Isosceles Triangle Theorem.
angle APC is a right angle.
triangle APC is a right triangle.
I have no clue how to approach this, please help!
Found 2 solutions by solver91311, DrBeeee:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Good start. Triangle APC is a right triangle. Using similar logic, show that Triangle APB is also right. Then since AC congruent to AB and AP congruent to AP by reflexive equality, the two right triangles are congruent by the Hypotenuse-Leg Postulate. Then PB is congruent to PC by CPCT and finally triangle BPC is isosceles by definition. What about the case where BC is also congruent to BP and CP? No problem, the set of isosceles triangles does not exclude equilateral triangles. By the way, you can get rid of that bit about angles ABC and ACB being congruent. They are, but nobody cares and it doesn't help you get where you are going.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
Answer by DrBeeee(684)  (Show Source):
You can put this solution on YOUR website! Continue your correct reasoning.
Since AB = AC (isosceles triangle ABC))
AP = AP Identity
Your right triangles
BPA and CPA are congruent
Then
BP = CP legs of congruent triangles
Therefore
Triangle BPC is an isosceles triangle (two legs are equal)
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