Circle Intersecting Triangle

[cbenk121]

Which of the following lists the number of points at which a circle can intersect a triangle?

(A) 2 and 6 only
(B) 2, 4, and 6 only
(C) 1, 2, 3, and 6 only
(D) 1, 2, 3, 4, and 6 only
(E) 1, 2, 3, 4, 5, and 6

OA: E

[palvarez]

1 when one of its sides is a tangent
2 when one of its sides is a secant
3. when one of its sides is a tangent and the other being a secant
4. when two of its sides are secants.
5. when two of its sides are secants, the other being a tangent
6. when all three sides are secants.

tangent = touches once
secants = intersects twice

[Testluv]

Which of the following lists the number of points at which a circle can intersect a triangle?

(A) 2 and 6 only
(B) 2, 4, and 6 only
(C) 1, 2, 3, and 6 only
(D) 1, 2, 3, 4, and 6 only
(E) 1, 2, 3, 4, 5, and 6

OA: E

Hi cbenk121,

We only need two things here: One, just one geometry fact, and, two, good strategy in working with the answer choices.

Geometry fact: When a line is tangent to--just barely touches--a curve, it intersects the curve
When you drew the triangle, you likely saw that you could draw a circle outside of (and beside) the triangle with one of the lines of the triangle just barely touching the curve of the circle. That is, you could draw one of the lines of the triangle tangentially to the curve of the circle. That would be one point of intersection.

If we didn't know that tangency counted as intersection, then choice B is mighty tempting because you would only be able to draw the circles intersecting the triangle in an even number of ways (2, 4, 6). (ie, "secant" as Palvarez points out).

But, the moment we remember (or divine) that tangency counted as intersection, we know the circle can intersect the triangle in an odd number of ways as well. (As soon as you drew the intersection with one point, you would be able to tell that the 6 and 4 triangles could be modified to 5 and 3).

Note that when we are dealing with two lines, intersection is just the lines crossing each other. (They can't be "tangent").

[Abdulla]

It is an interesting question. Can someone draw it pls? I tired to draw it, but it does not work.

[cbenk121]

Which of the following lists the number of points at which a circle can intersect a triangle?

(A) 2 and 6 only
(B) 2, 4, and 6 only
(C) 1, 2, 3, and 6 only
(D) 1, 2, 3, 4, and 6 only
(E) 1, 2, 3, 4, 5, and 6

OA: E

Hi cbenk121,

We only need two things here: One, just one geometry fact, and, two, good strategy in working with the answer choices.

Geometry fact: When a line is tangent to--just barely touches--a curve, it intersects the curve
When you drew the triangle, you likely saw that you could draw a circle outside of (and beside) the triangle with one of the lines of the triangle just barely touching the curve of the circle. That is, you could draw one of the lines of the triangle tangentially to the curve of the circle. That would be one point of intersection.

If we didn't know that tangency counted as intersection, then choice B is mighty tempting because you would only be able to draw the circles intersecting the triangle in an even number of ways (2, 4, 6). (ie, "secant" as Palvarez points out).

But, the moment we remember (or divine) that tangency counted as intersection, we know the circle can intersect the triangle in an odd number of ways as well. (As soon as you drew the intersection with one point, you would be able to tell that the 6 and 4 triangles could be modified to 5 and 3).

Note that when we are dealing with two lines, intersection is just the lines crossing each other. (They can't be "tangent").

It all came down to one question: does tangent mean an intersection? I guess it didn't (B), and consequently got it wrong/learned something in the process. This is why practice problems are good, see these little quirks .