# Points A and B lie in the same plane How many isosceles right triangles in the plane have A and B as vertices?

Update:

(A)Three

(B)Four

(C) Five

(D)Six

(E)Infinitely many

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• Anonymous
2 years ago

An isosceles right triangle has angles 45º-45º-90º.

If AB is a hypotenuse there are 2 possible triangles:

. .P

A. . B

. .Q

The triangles are ABP and ABQ with the 90º angles at P and Q.

If AB is one of the equal legs, there are another 4 possible triangles:

. . . R

.

T. . . . . . B

.

. . A. . . . . . S

.

. . . . . . . U

The triangles are

ABR with the 90º angle at B

ABS with the 90º angle at B

ABT with the 90º angle at A

ABU with the 90º angle at A

Total 2+4 = 6. Answer D.

(Diagrams not accurately drawn.)

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• D. Six

Drag points A and B to the required positions. • thank you so much

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• I get six. two each for when a point is the vertex of the right angle, and two each where neither point is the vertex of a right angle. Really, what you have is three possible isoceles right triangles that include those two points lying on one side of the line defined by those points, and the mirror image of those three triangles (the same thing but on the other side of the line).

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