12 multiple-choice questions, progressively harder.
Triangles ABCABCABC and DEFDEFDEF have AB=DE=7AB=DE=7AB=DE=7, BC=EF=9BC=EF=9BC=EF=9, and CA=FD=12CA=FD=12CA=FD=12. Which method proves their congruence most directly from these data?
Solution
Correct answer: B
The three corresponding side equalities are already given, so SSS proves congruence without introducing coordinates. The lengths form nondegenerate triangles because 7+9>127+9>127+9>12.
A coordinate proof could calculate side lengths if coordinates were given, but here that would add work. No right angle was supplied, and equal perimeters alone do not establish congruence.
A nondegenerate rectangle is placed with A(0,0), B(a,0), C(a,b), D(0,b), with a,b positive. Which is the squared length of diagonal BD?
Correct answer: D
From B to D, the coordinate changes are -a and b.
To prove a statement about every scalene triangle, which coordinate choice needs correction?
Correct answer: A
The axes meet at a right angle. Putting both sides on them requires the triangle to be right.
A scalene triangle may be right, acute, or obtuse, so that placement does not cover the whole claim.
A triangle has A(0,0), B(10,0), C(2,8). M and N are the midpoints of AC and BC. What is MN?
Correct answer: C
The midpoints are M(1,4) and N(6,4).
Their common height also shows that MN is parallel to AB.
What is the centroid of the triangle with vertices (1,2), (7,-4), and (10,8)?
Average three coordinates in each direction.
A convex quadrilateral has A(1,1), B(6,2), C(8,7), D(3,6) in boundary order. Both diagonal midpoints are (9/2,4). What does this calculation alone establish?
The same point is the midpoint of each diagonal, so the diagonals bisect each other.
The diagonal-bisection test establishes a parallelogram. It supplies no side equality or right-angle test.
The boundary vertices of a rectangle are (-3,2), (4,2), (4,6), and (-3,6). What is its perimeter?
Its side lengths are 7 and 4 units.
A triangle has vertices (-2,-1), (6,-1), and (1,4). What is its area?
The horizontal base has length 8. Its line is at height -1, and the opposite vertex has height 4, so the perpendicular height is 5.
A simple pentagon has vertices (0,0), (6,0), (6,3), (4,5), and (0,3) in boundary order. Which decomposition gives its area?
The horizontal segment from (0,3) to (6,3) lies inside the pentagon.
Below it is the 6 by 3 rectangle; above it is a triangle with base 6 and perpendicular height 2. The pieces fill the polygon without overlapping interiors.
A coordinate calculation verifies a claim for one rectangle of width 9 and height 2. What is still needed to prove the claim for every rectangle?
A fixed width and height describe one instance.
Arbitrary positive side lengths keep all nondegenerate rectangles in the argument; another picture of the same instance does not.
A square is verified by coordinates. Which additional classification must also apply under the inclusive definitions?
A square has four right angles, so it meets the rectangle definition.
It also has four congruent sides, so it meets the rhombus definition.
A polygon is divided by an internal diagonal to calculate area. How is this diagonal treated in the outside perimeter?
An internal diagonal is not part of the outside boundary.
It helps identify area pieces, while perimeter follows only the exposed edges.
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