C972 College Geometry
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Free C972 College Geometry Questions
In a trig identities lesson without calculations, students use unit circle diagrams. sin²θ + cos²θ =1 represents what property?
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Pythagorean identity on unit circle
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Angle addition
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Double angle
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Law of sines
Explanation
On the unit circle, any point corresponding to angle θ has coordinates (cosθ, sinθ). By the Pythagorean theorem applied to the radius of the unit circle (r = 1), the relationship sin2 θ + cos2 θ= 1 holds. This is the Pythagorean identity, a fundamental property of trigonometric functions derived geometrically from the unit circle.
What is the midpoint of segment joining (-2,5) and (4,-3)?
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(1,1)
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((-2+4)/2, (5-3)/2) = (1,1)
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(1,4)
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(3,1)
Explanation
A teacher introduces 3D geometry with everyday objects. The dihedral angle is best described as the angle between what?
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Two intersecting planes
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Edge and face
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Vertex and edge
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Volume measures
Explanation
A dihedral angle is the angle formed between two intersecting planes in three-dimensional space. It is measured along the line where the planes meet, which is their common edge. Using everyday objects helps students visualize this concept by observing angles formed by, for example, book covers or boxes.
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.
What is the area of the circle sector in Circle Sector C3 rounded to the nearest hundredths?
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18 units
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339.29 square units
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3.14
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120º
Explanation
The area of a sector of a circle is calculated using the formula A = (θ/360)*π r2, where θ is the central angle in degrees and r is the radius. This formula gives the portion of the circle’s area that the sector occupies. After substituting the given values for θ and r and performing the calculation, the area is found to be 339.29 square units when rounded to the nearest hundredths. Units must be squared because area is a two-dimensional measurement.
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 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.
A teacher uses Geometer's Sketchpad to demonstrate circle theorems. Which theorem shows that angles in the same segment are equal?
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Inscribed angle theorem
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Alternate segment theorem
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Angle at center is twice inscribed
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Tangent perpendicular to radius
Explanation
The Inscribed Angle Theorem states that angles subtended by the same chord in the same segment of a circle are equal. This means any two angles lying on the circumference and sharing the same chord as their base will have identical measures. This theorem is fundamental in circle geometry and is often used in proofs involving cyclic quadrilaterals and other circle properties.
In a secondary classroom, to teach Pythagorean theorem, what real-world scenario works best?
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Diagonal of a TV screen
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Shadow of a ladder on ground
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Both A and B
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Area of square
Explanation
The Pythagorean theorem is most effectively taught through real-world scenarios involving right triangles. Both the diagonal of a TV screen and the shadow of a ladder on the ground provide concrete examples where the relationship a2+b2=c2 applies. These situations help students visualize and calculate the lengths of sides in right triangles, making the theorem meaningful and applicable beyond abstract exercises.
In a secondary math classroom, a teacher asks students to construct a perpendicular bisector of a segment using a compass and straightedge. Which of the following steps is the first action after drawing the segment AB?
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Draw arcs from point A with radius greater than half AB
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Mark the midpoint of AB
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Draw arcs from the midpoint
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Connect the intersection points
Explanation
To construct a perpendicular bisector of a segment using a compass and straightedge, after drawing segment AB, the next step is to draw arcs from point A with a radius greater than half the length of AB. These arcs will intersect with arcs drawn from point B, creating two intersection points. Connecting these intersection points produces the perpendicular bisector, which passes through the midpoint of AB. Drawing arcs first ensures that the bisector is accurate without needing to know the midpoint in advance.
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