12 multiple-choice questions, progressively harder.
D,E lie inside AB,AC. Suppose
What value of x makes DE parallel to BC?
Solution
Correct answer: D
Use the converse with equal part-to-part ratios.
The four parts are positive at this value, so the interior hypotheses are consistent.
In the converse proof, two interior points on AC have the same ratio of first part to leftover part. Why must they coincide?
Correct answer: A
Writing the first parts as x,y and the whole as L gives x/(L-x)=y/(L-y). Cross-multiplication cancels the common xy term, giving xL=yL and hence x=y; equal distances on the same ray identify the point.
D,E are midpoints of AB,AC. G,H are midpoints of AD,AE. If BC=58BC=58BC=58, what is GH?
Correct answer: C
The first midpoint cut gives DE half of BC. The next gives GH half of DE.
D,E,F are the midpoints of AB,AC,BC. What is the factor from central triangle DEF to outer triangle CBA under their natural correspondence?
Correct answer: B
Central side DE is half of outer side CB, and the other corresponding pairs have the same relation. Going from the central triangle to the outer triangle therefore doubles the lengths.
Two transversals cross at O between the first and second parallel cross-lines. O lies between A,B and between P,Q. Which operations give the intercepted lengths AB and PQ from their endpoint distances to O?
Each intercepted interval crosses O, so it is the sum of the two endpoint distances. A common vertex-distance factor survives that addition. Differences apply only when both endpoints lie on the same ray from O.
A parallel cross-line passes through the intersection O of the two transversals. How is that case handled in the proportional-interval argument?
No degenerate triangle is needed. The cross-line through O contributes zero vertex distances, and the same addition or subtraction with the other endpoint distances preserves the established factor.
Why does the parallel-transversal case use a shared diagonal and ASA congruence?
The transversals do not meet, so distances from their intersection are unavailable. Each strip diagonal produces two equal-angle pairs from the parallel lines and a shared included side, allowing ASA congruence.
The compass opening is changed while auxiliary steps are marked. What is the precise loss?
Changing openings removes the equal-length premise. A special choice might still accidentally give some desired ratio, but the equal-parts guarantee is lost.
The auxiliary point U is copied at distance a from A, and V is copied a further distance b along the same ray. UX is constructed parallel to VB. Which ratio follows for the interior point X on AB?
Apply side-splitter in triangle AVB. The consecutive auxiliary parts are
, so AX/XB equals a/b. The whole-side fraction would instead be a/(a+b).
Which statement correctly limits the supplied-segment construction?
The construction starts from actual supplied segments. Copying their lengths and drawing parallels transfers their ratio, without asserting that every numerical real length can be constructed from scratch.
Three parallel cross-lines meet two parallel transversals. In the first strip, one transversal intercept is 6.5 units. In the second strip it is 10 units. What is the first-strip intercept divided by the second-strip intercept on the other transversal?
Parallel transversals have equal corresponding intercepts in each strip, by the ASA argument.
A parallel cut gives AD/AB=3/8.AD/AB=3/8.AD/AB=3/8. Which part-to-part ratio is forced?
The whole has eight ratio parts. AD uses three and DB uses the remaining five, so the part-to-part ratio is 3:5 rather than 3:8.
Reset this practice set?
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