12 multiple-choice questions, progressively harder.
A convex quadrilateral has vertices A(-2,-1), B(4,2), C(2,6), D(-4,3) in boundary order. What is its most specific classification among these choices?
Solution
Correct answer: C
Both diagonal midpoints are (0,5/2), so it is a parallelogram.
An adjacent right angle makes it a rectangle. Adjacent squared lengths are 45 and 20, so it is not a square.
A convex quadrilateral has A(-5,1), B(1,5), C(7,1), D(1,-3). Which exact calculation supports a rhombus classification after its diagonals have a common midpoint?
AB2=52AB^2=52AB2=52
BC2=52BC^2=52BC2=52
Correct answer: A
The adjacent changes are (6,4) and (6,-4).
Equal adjacent sides in the established parallelogram prove a rhombus. Its diagonals are 12 and 8, and the slope product is -4/9.
A(2,-3), B(8,-1), and C(11,7) are consecutive vertices of parallelogram ABCD. What is D?
Correct answer: D
The midpoint of AC is (13/2,2). The midpoint of BD must be the same.
A triangle has vertices A(-4,0), B(8,0), C(2,10). M and N are the midpoints of AC and BC. Which line contains MN?
Correct answer: B
Midpoints M(-1,5) and N(5,5) have the same second coordinate.
This is distinct from the base line at height zero.
A triangle has A(-6,4), B(3,-5), and centroid G(1,2). What is its third vertex C?
The three vertex coordinates sum to three times the centroid coordinates.
The side AB has slope -1, whereas AC has slope 1/4, so the recovered triangle is nondegenerate.
A simple hexagon has boundary vertices (-1,-2), (7,-2), (7,1), (2,1), (2,5), and (-1,5). What is its area?
The containing rectangle has width 8 and height 7. The upper-right missing rectangle has width 5 and height 4.
A simple quadrilateral has boundary vertices (0,0), (5,0), (5,3), and (1,3). What is its exact perimeter?
The horizontal sides have lengths 5 and 4, and the right side has length 3. The remaining side has gaps 1 and 3.
A map uses meters as coordinate units. A triangular planting region has vertices (2,1), (10,1), (6,7). A straight fence follows its whole boundary. What is the exact fence length?
The base is 8 meters. Each slanted side has horizontal gap 4 and vertical gap 6, so each is twice the square root of 13.
For a rectangle with A(0,0), B(a,0), C(a,b), D(0,b), a,b positive, which observation proves diagonal bisection?
Averaging A with C gives (a/2,b/2). Averaging B with D gives the same point.
Thus each diagonal has the same midpoint; this is precisely diagonal bisection. Their slopes instead have opposite signs.
A nondegenerate triangle has A(0,0), B(2a,0), C(2b,2c), with a,c positive. Midpoints M of AC and N of BC are joined. Which condition on b is needed for the midsegment proof?
The midpoints have coordinates (b,c) and (a+b,c).
Their common positive height ensures a line distinct from AB, whatever the real value of b.
Triangle ABC has A(-2,3), B(4,1), C(7,8). D is the midpoint of BC. Which ordered ratio locates the centroid G internally on AD?
The centroid occupies two of the three parts when starting from the vertex.
Thus AG is twice GD, while AG is two thirds of the whole AD. The direction and denominator must name the requested segments.
A proof aims to show that every parallelogram has diagonals that bisect each other. Which starting coordinate model includes all nondegenerate parallelograms after choosing axes?
A horizontal side can have arbitrary positive length a. The adjacent side can have any horizontal change b and any positive vertical change c after orienting the axes.
The other vertices keep the same side changes. Forcing b to zero would restrict the proof to rectangles.
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