C881 Geometry for Secondary Mathematics Teaching
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Free C881 Geometry for Secondary Mathematics Teaching Questions
The Brocard points of a triangle have property:
- A. Centroid
- B. Isogonal conjugates
- C. Angles to sides are equal (ω)
- D. Only in equilateral
Explanation
Brocard points of a triangle are two special points inside a triangle such that the angles from each vertex to the point are equal, called the Brocard angle (\omega). These points are not necessarily the centroid and exist in all non-degenerate triangles. They are not limited to equilateral triangles, though in that case the Brocard points coincide with other notable centers.
Correct Answer:
Angles to sides are equal (ω)
The power of point P outside circle with secants PAB and PCD is:
- A. Chord lengths equal
- B. Tangent only
- C. PA × PB = PC × PD
- D. PA² = PB²
Explanation
The power of a point theorem states that for a point P outside a circle, the product of the lengths of the segments of one secant equals that of another secant drawn from the same point: (PA × PB = PC × PD). This theorem generalizes the tangent-secant relationship and provides a tool for solving many circle geometry problems. Other options do not capture the correct relationship for two secants intersecting outside a circle.
Correct Answer:
PA × PB = PC × PD
A dilation centered at (2,3) with k = –1 maps (5,7) to:
- A. (2,3)
- B. (–5,–7)
- C. (–1,–1)
- D. (8,11)
Explanation
A dilation with scale factor (k) centered at ((h,k)) transforms a point ((x,y)) according to: (x',y') = (h + k(x-h), k + k(y-k)). Substituting the given values: x' = 2 + (-1)(5-2) = 2 - 3 = -1 y' = 3 + (-1)(7-3) = 3 - 4 = -1 Thus, the image of (5,7) under this dilation is (–1,–1).
Correct Answer:
(–1,–1)
The orthocentric system consists of:
- A. Midpoints only
- B. Incenter and excenters
- C. Only four points in right triangles
- D. Orthocenter, centroid, circumcenter, and reflection points
Explanation
An orthocentric system in a triangle refers to the set of points formed by the orthocenter, centroid, circumcenter, and reflections of the orthocenter over the sides. These points exhibit special collinear and symmetry properties and are central to the study of triangle centers and their interrelations. This system is more comprehensive than just midpoints, incenters, or a subset in right triangles.
Correct Answer:
Orthocenter, centroid, circumcenter, and reflection points
The polar equation r = 6 cos θ represents:
- A. Spiral
- B. Limacon
- C. Circle with diameter from (0,0) to (3,0)
- D. Cardioid
Explanation
The polar equation r = a cos θ represents a circle with diameter along the polar axis. Here, r = 6 cos θ describes a circle with diameter 6, centered at (3,0) in Cartesian coordinates. It is not a spiral, limacon, or cardioid, which have different polar forms.
Correct Answer:
Circle with diameter from (0,0) to (3,0)
The circumcenter is the intersection point of:
- A. Altitudes
- B. Medians
- C. Angle bisectors
- D. Perpendicular bisectors
Explanation
The circumcenter of a triangle is the point where the perpendicular bisectors of its sides intersect. It is equidistant from all three vertices and serves as the center of the circumscribed circle. This is distinct from medians, altitudes, or angle bisectors, which intersect at other special points like the centroid, orthocenter, or incenter.
Correct Answer:
Perpendicular bisectors
The vector equation of a line through (3,–2) parallel to <4,5> is:
- A. r = <3,–2> × <4,5>
- B. r = <3,–2> + t<–4,–5>
- C. r = <3,–2> + t<4,5>
- D. r = <4,5> + t<3,–2>
Explanation
The vector equation of a line through a point ((x0, y0)) in the direction of a vector vec{v} = < vx, vy > is given by vec{r} = < x0, y0 > + t vec{v} , where t is a scalar parameter. Here, the line passes through (3,–2) with direction vector <4,5>, giving vec{r} = < 3, –2 > + t < 4, 5 >. Other options either reverse the direction, use the wrong operation, or place the point and vector incorrectly.
Correct Answer:
r = <3,–2> + t<4,5>
The coordinates of a triangle's vertices are A(1,2), B(3,4), C(5,1). What is the length of side AB?
- 2√2
- √8
- 4
- √10
Explanation
Correct Answer:
2√2
Teaching exterior angle theorem, best visual:
- Extend side and compare to remote interiors.
- Use only interior sums.
- Calculate with sine law.
- Ignore triangles for quadrilaterals.
Explanation
The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two remote interior angles. The most effective visual approach is to extend one side of the triangle to form an exterior angle and directly compare it to the two non-adjacent interior angles. This helps students see the relationship clearly and reinforces the concept through geometric visualization. Using only interior sums or sine calculations does not provide the same intuitive understanding, and ignoring triangles is irrelevant.
Correct Answer:
Extend side and compare to remote interiors.
In a classroom debate, students argue if a rectangle is a parallelogram. Which definition supports yes?
- Opposite sides parallel and equal.
- All angles 90°.
- Diagonals bisect each other.
- Four right angles only.
Explanation
A rectangle is a special type of parallelogram because it satisfies the defining property of parallelograms: opposite sides are both parallel and equal in length. While a rectangle also has four right angles and bisecting diagonals, the essential characteristic that confirms it as a parallelogram is the parallelism and equality of opposite sides. The other properties describe additional features of rectangles but are not the foundational definition that classifies it as a parallelogram.
Correct Answer:
Opposite sides parallel and equal.
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