C972 College Geometry
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Free C972 College Geometry Questions
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.
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.
What are adjacent angles?
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Two angles whose measures add to 90 degrees
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Two nonadjacent angles formed by intersecting lines
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Two angles whose measures add to 180 degrees
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Angles that share a vertex and a common side
Explanation
Adjacent angles are two angles that sit next to each other, sharing both a common vertex and a common side. They do not overlap and are positioned so that one angle is immediately beside the other. This relationship is based purely on their placement, not on their measures. Although some adjacent angles can also be complementary or supplementary, adjacency itself only requires the shared vertex and side
A teacher uses a clinometer to measure tree height. If angle of elevation is 30° from 50 ft away, height ≈ ?
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50tan(30°) ≈ 28.9 ft
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50/sin(30°)
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50cos(30°)
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50 + eye height
Explanation
To find the height of a tree using the angle of elevation and distance from the tree, the tangent function is applied: tan θ = opposite/adjacent. Here, the opposite side is the tree height and the adjacent side is 50 ft. So, height = 50tan(30°) ≈ 28.9 ft. This calculation uses basic trigonometry for right triangles.
Ceva's theorem in trig form uses sine ratios of angles for what condition?
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Concurrency of lines from vertices
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Parallelism
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Similarity
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Collinearity
Explanation
Ceva’s theorem provides a criterion for three cevians (lines from vertices to opposite sides) to be concurrent. In its trigonometric form, it expresses the condition using sine ratios of certain angles in the triangle. This allows students to determine concurrency of cevians using angle measures instead of only segment ratios.
Which statement is an axiom in Hilbert's foundations of geometry?
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Between any two points there is a third point
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A plane contains at least three non-collinear points
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All straight angles are equal
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Incidence: a line lies on a plane
Explanation
Hilbert's axioms formalize geometry rigorously. Incidence axioms describe how points, lines, and planes relate. One of the incidence axioms states that a line lies on a plane, defining a fundamental relationship in space. This statement is an accepted postulate without proof and forms the basis for constructing other geometric properties
What tool is used to create the sector within the circle in Circle Sector C2?
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about tool
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between Two tool
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around tool
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with Given tool
Explanation
In geometric drawing software, the "between Two tool" is used to create a sector of a circle because it allows the user to define a sector by selecting two points on the circumference or two bounding radii. This tool effectively captures the portion of the circle between those two defining points, forming the sector. Other tools such as "about," "around," or "with Given" are used for different constructions, like rotations, full circles, or measurements, and do not directly create sectors.
What is the area of a regular hexagon with side length 4?
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(3√3/2)16 = 24√3
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6(√3/4)16 = 24√3
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(3√34²)/2
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644*sin(60°)
Explanation
The area of a regular hexagon can be calculated by dividing it into six equilateral triangles. The area of one equilateral triangle with side length s is (√3/4)s2. Multiplying by six gives 6×(√3/4)×42 = 6 × (√3/4) × 16 = 24√3. This formula efficiently calculates the total area of the hexagon using its side length.
Sine of 150°?
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1/2
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sin(180°-30°) = sin30°
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√3/2
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-1/2
Explanation
Using the sine identity for angles in the second quadrant: (180° - θ) = sin θ. Here, 150° = 180° - 30°, so sin 150° = sin 30° = 1/2. This uses the property of sine that it is positive in the second quadrant and symmetric about 90°.
What are the coordinates of point D in Triangle DEF?
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(6,1)
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(4,0)
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(0,0)
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(3,2)
Explanation
The coordinates of a point are given as an ordered pair (x, y) on a coordinate plane. Point D in Triangle DEF is identified by its specific position relative to the axes. Based on the provided information, point D is located at (3,2), which specifies its horizontal distance from the y-axis and vertical distance from the x-axis. Correctly identifying coordinates is crucial for plotting figures, performing transformations, and solving geometric problems.
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