12 multiple-choice questions, progressively harder.
You have drawn the perpendicular bisector of BC with compass arcs in a scalene triangle ABC. What is the next drawing step for constructing the median from A?
Solution
Correct answer: B
The perpendicular bisector meets BC at its midpoint D. Joining the opposite vertex A to D constructs median AD. In a scalene triangle, the perpendicular bisector does not pass through A, so the line used to locate D is a different line from the median.
Side QR is 26 cm long. M is its midpoint, and PM is a median of triangle PQR. What is QM?
Correct answer: D
A midpoint splits the side into two equal parts.
Which intersection defines the centroid of a nondegenerate triangle?
Correct answer: A
The centroid is the common point of the three medians. The perpendicular bisectors and internal angle bisectors define other centers.
G is the centroid on median AD, where A is the vertex and D the opposite side midpoint. What is the ratio AG to GD?
Correct answer: C
The centroid divides each median into two ratio parts next to the vertex and one next to the side midpoint. The requested order is vertex part to midpoint part.
Median AD has length 27 cm. G is the centroid. What is AG?
The vertex part is two thirds of the whole median.
G is the centroid on median BE. If GE is 6 cm, what is BG?
GE is the portion next to the side midpoint E. The vertex portion is twice as long.
Where is the centroid of a nondegenerate obtuse triangle?
The medians cross strictly inside a nondegenerate triangle. This remains true for obtuse triangles.
Which property is required by the definition of a median?
The definition concerns the two endpoints of a segment: a vertex and the opposite midpoint. Perpendicularity and angle bisection are additional properties in some triangles.
When constructing the midpoint of BC using equal-radius arcs centered at B and C, why choose a radius greater than half of BC?
With equal radii greater than half of BC, the circles meet at two distinct points on opposite sides of BC. Their joining line constructs its perpendicular bisector.
What fraction of a median is the segment from the centroid to the opposite side midpoint?
The whole median has three ratio parts. Exactly one lies between the centroid and the side midpoint.
For which ideal object does the centroid balance statement in this lesson apply?
The stated balance interpretation assumes material distributed uniformly through a thin triangular sheet. An outline alone does not specify that mass distribution.
D, E, F are the midpoints of BC, CA, AB respectively. Which list contains the three medians?
Join each vertex to the midpoint of its opposite side: A to D, B to E, and C to F.
Reset this practice set?
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