C972 College Geometry
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Free C972 College Geometry Questions
For a circle with radius 5 cm, what is the length of a 72° arc?
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(72/360)2π5 = 2π cm
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72π/5 cm
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5π cm
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(72/360)π25 = (1/5)π*25 = 5π cm
Explanation
In teaching coordinate proofs, to prove a quadrilateral is a parallelogram, what vector condition must hold?
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AB = DC and AD = BC
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Diagonals bisect
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Midpoints of diagonals coincide
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All of the above
Explanation
A quadrilateral is a parallelogram if any of several equivalent conditions are satisfied. Vectors provide a convenient method: opposite sides must be equal (AB = DC and AD = BC). Additionally, the diagonals of a parallelogram bisect each other, meaning the midpoints of the diagonals coincide. Since all these conditions are equivalent and confirm that the quadrilateral is a parallelogram, "all of the above" is the correct choice.
What is the definition of coplanar points?
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Points on the same line
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Points on the same plane
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Points that intersect
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Points on different planes
Explanation
Coplanar points are points that lie on the same plane. A plane is a flat surface that extends indefinitely, and any group of points that can be placed together on such a surface are considered coplanar. This concept applies even if the points do not lie in a straight line or intersect; they simply need to share the same plane. Coplanarity is broader than collinearity, since many different arrangements of points can lie in a single plane.
Equation of circle center (2,-1), r=√5?
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(x-2)² + (y+1)² = 5
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x² + y² = 5
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(x+2)² + (y-1)² = 5
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(x-2)² + (y-1)² = 25
Explanation
The equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r2. Here, the center is (2, -1) and the radius is √5. Squaring the radius gives (√)2 = 5. Substituting h = 2 and k = -1 into the formula results in (x - 2)2 + (y + 1)2 = 5.
What is the slope of the line perpendicular to y=2x+1?
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-1/2
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2
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1/2
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-2
Explanation
The slope of a line perpendicular to another line is the negative reciprocal of the original line’s slope. The given line has the equation y = 2x + 1, where the slope is 2. The negative reciprocal of 2 is -1/2. Therefore, any line perpendicular to y = 2x + 1 will have a slope of -1/2, ensuring the lines intersect at a right angle.
What are Triangle ABC's point B's coordinates?
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(4,4)
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(0,0)
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(6,1)
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(3,6)
Explanation
The coordinates of a point are expressed as an ordered pair (x, y) that identifies its location on a coordinate plane. Point B in Triangle ABC is located at (6,1), indicating it is 6 units along the x-axis and 1 unit along the y-axis. Knowing the coordinates is essential for plotting triangles, performing transformations, and calculating distances or slopes between points.
To teach reflection symmetry, a teacher folds a figure. What must coincide?
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Corresponding points
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Line of reflection is perpendicular bisector
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Angles preserved
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All of the above
Explanation
Reflection symmetry occurs when a figure can be folded along a line so that one half maps exactly onto the other. For this to happen, corresponding points must coincide, the line of reflection must act as the perpendicular bisector of segments joining corresponding points, and angles must be preserved to maintain congruence. Therefore, all these conditions are required to demonstrate reflection symmetry
In discussing axiomatic systems, a class debates undefined terms. Which is typically left undefined in Euclidean geometry?
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Point, line, plane
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Distance
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Angle
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Congruence
Explanation
In Euclidean geometry, certain basic concepts are left undefined to form the foundation of the axiomatic system. These include point, line, and plane. By not defining them, other definitions, postulates, and theorems can be built logically from these fundamental concepts, ensuring a rigorous and consistent geometric framework.
What is the image of Triangle ABC under the translation T (2,2)?
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A sector
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A line
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A new triangle
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A circle
Explanation
A translation in geometry moves every point of a figure the same distance in the same direction. For the translation T(2,2), each vertex of Triangle ABC is shifted 2 units to the right and 2 units up. This preserves the shape and size of the triangle, producing a new triangle that is congruent to the original. Translations do not change the type of geometric figure, so the image remains a triangle.
Which quadrilateral has diagonals that are perpendicular bisectors of each other?
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Rhombus
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Rectangle
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Square
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Kite
Explanation
In a rhombus, the diagonals intersect at right angles and bisect each other. This property distinguishes it from other quadrilaterals. While a square also has perpendicular bisecting diagonals, the defining property for a rhombus (all sides equal) ensures that the diagonals are perpendicular regardless of angles. A rectangle has bisecting diagonals but they are not perpendicular, and a kite has perpendicular diagonals but only one diagonal is bisected.
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