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Geometry / Mr. Hansen |
Name:
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Big Quiz on Locus and Concurrency
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Part I. |
Instructions: Print the capital letter of the best choice in the blank provided. Points may be deducted if you do not print a capital letter. There is no partial credit. Each question is worth 4 points. |
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__1. |
For a thin triangular region of uniform density, the centroid is always the same as the . . . |
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(A) incenter (B) circumcenter (C) center of gravity |
(D) orthocenter (E) [none of these] |
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__2. |
For an isosceles triangle, the incenter and the circumcenter are always . . . |
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(A) collinear (B) coincident (i.e., at the same point) (C) on opposite sides of the orthocenter |
(D) both (A) and (C) (E) both (B) and (C) |
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__3. |
The perpendicular bisectors of any triangle always coincide at a point that is . . . |
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(A) equidistant from the vertices (B) equidistant from the sides (C) inside the triangle |
(D) outside the triangle (E) on the longest side of the triangle |
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__4. |
Where is the orthocenter of an equilateral triangle located along a segment that starts at a vertex and terminates at the opposite side of the triangle? |
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(A) one-third of the way (i.e., closer to the vertex than to the opposite side) |
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(B) two-thirds of the way (i.e., closer to the opposite side than to the vertex) |
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(C) halfway between the vertex and the opposite side |
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(D) location varies depending on the size of the equilateral triangle |
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(E) [insufficient information to answer] |
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__5. |
What is the locus of points in a plane that are at a fixed positive distance from point P? |
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(A) a point (B) a line (C) a line segment |
(D) a circle (E) a sphere |
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__6. |
What is the locus of points in 3-space that are no more than 2 cm away from point Q? |
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(A) a point (B) a line segment without its endpoints (C) null set |
(D) a filled sphere (i.e., a sphere in union with its boundary) (E) either (C) or (D) |
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__7. |
What is the locus of points in 3-space that are equidistant from two fixed points (P and Q) and equidistant from two other fixed points (R and S)? |
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(A) null set (B) a single point (C) a line |
(D) a plane (E) any of these
are possible |
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__8. |
Why is it harder to sketch the orthocenter of an obtuse triangle than it is to sketch the orthocenter of an acute triangle? |
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(A) The sides of the obtuse triangle must be extended. |
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(B) The obtuse triangle may be isosceles. |
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(C) The obtuse triangle may already have a right angle in it. |
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(D) The orthocenter of an obtuse triangle lies on one of the sides. |
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(E) None of these, since it is no harder to sketch the orthocenter of an obtuse triangle. |
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__9. |
Locus 1 is a straight line. Locus 2 is a straight line. The compound locus that satisfies both sets of requirements (1 and 2) is . . . |
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(A) null set (B) a single point (C) a line |
(D) (A), (B), or (C) (E) [none of these] |
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Part II: Short answer (8 pts. each) |
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10. |
Define the word “locus” as used in geometry. |
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11. |
Sketch the possibilities for the locus of points in a plane that are equidistant from two points (A and B) and lie on a given line. |
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12. |
Sketch a right triangle that is not an isosceles triangle, together with its incenter and circumcenter clearly marked as C and I. Make a fairly large diagram. |
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