C972 College Geometry
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Free C972 College Geometry Questions
In teaching similarities, students compare photos taken from different distances. What type of transformation models camera zoom?
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Dilation
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Translation
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Reflection
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Shear
Explanation
Dilation is a transformation that enlarges or reduces a figure proportionally from a fixed center, preserving shape but changing size. Camera zoom works similarly: objects appear larger or smaller depending on distance, but their shapes remain similar. This demonstrates the concept of similarity through scaling.
In 3D geometry, what is the shortest path on a cube surface from one vertex to opposite vertex?
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Unfold to net, straight line
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√(a² + (2a)²) = a√5
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3a
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Both A and B
Explanation
To find the shortest path along the surface of a cube from one vertex to the opposite vertex, the cube can be “unfolded” into a net. On the net, the path becomes a straight line across connected faces. For a cube of side length a, unfolding appropriately gives a right triangle with sides a and 2a, so the straight-line distance is √(a2 + (2a)2) = a√5. Both the method (unfolding) and the calculation are valid, making both correct.
To teach the golden rectangle, a teacher shows examples in art and nature. Which famous structure approximates the golden ratio?
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Parthenon
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Eiffel Tower
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Sydney Opera House
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Great Pyramid
Explanation
The golden ratio, approximately 1.618, appears in proportions of various famous structures. The Parthenon in Greece is a well-known example, where its façade and elements approximate the golden rectangle. This provides a tangible real-world illustration of mathematical concepts in art and architecture, helping students connect geometry with historical and aesthetic examples.
For a central angle of 120° in a circle of radius 7, what is the area of the sector?
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(120/360)π49 = (1/3)49π = 49π/3
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120π7
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(1/2)77sin(120°)
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120/360 * 2π7
Explanation
Which axiom in Euclidean geometry states that two distinct points determine a unique line?
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If two points lie on a line, the line contains them
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A line contains at least two points
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Through any two points, there is exactly one line
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All right angles are equal
Explanation
The Euclidean axiom stating that two distinct points determine a unique line is the fundamental principle that through any two points, exactly one straight line can be drawn. This axiom underpins the definition of lines and segments in Euclidean geometry and ensures the uniqueness of the line connecting any pair of points. It is a cornerstone of geometric reasoning, providing the basis for constructing figures and proving relationships between points and lines.
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.
Teaching locus concepts, a teacher asks for points equidistant from two fixed points. Students describe it as the set forming what?
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Perpendicular bisector line
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Circle around one point
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Parabola
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Two parallel lines
Explanation
The locus of points equidistant from two fixed points forms the perpendicular bisector of the segment connecting those points. Every point on this line is at the same distance from both fixed points, making it a straight line that divides the segment into two equal parts at a right angle. This geometric property is fundamental in both plane geometry and coordinate proofs.
Surface area of a cylinder with r =3, h =5 (exclude bases)?
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2π35 = 30π
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2π9 + 2π35
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π35
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2π3*(3+5)
Explanation
The lateral surface area of a cylinder (excluding bases) is calculated using the formula A=2πrh. Substituting r = 3 and h = 5: A = 2π(3)(5) = 30π. This formula represents the area of the curved surface of the cylinder only, without including the top and bottom circular bases.
Prove that the diagonals of a parallelogram bisect each other. Which is the key congruence
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SAS
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SSS
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ASA
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HL
Explanation
To prove that the diagonals of a parallelogram bisect each other, one typically examines the triangles formed by the diagonals. By showing that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, SAS (Side-Angle-Side) congruence is applied. This congruence confirms that the triangles are equal, which implies that the diagonals intersect at their midpoints, bisecting each other.
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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