12 multiple-choice questions, progressively harder.
AD is an altitude with D strictly inside BC. If ∠ABC=36∘\angle ABC=36^\circ∠ABC=36∘, what is ∠BAD\angle BAD∠BAD?
Solution
Correct answer: B
Triangle ABD is right at D, so its other two angles sum to 90 degrees.
Triangle ABC is right at C. Which side itself is the altitude from B?
Correct answer: C
BC starts at B and is perpendicular to the opposite side line AC. Its foot is C.
In the auxiliary construction for nondegenerate triangle ABC, A, B, C are the midpoints of QR, RP, PQ. Each auxiliary side is parallel to the opposite original side. Let X be the orthocenter of ABC. Which pair of equalities is guaranteed?
XP=XQXP=XQXP=XQ
XQ=XRXQ=XRXQ=XR
XA=XBXA=XBXA=XB
XB=XCXB=XCXB=XC
XP=PQXP=PQXP=PQ
XQ=QRXQ=QRXQ=QR
XA=ABXA=ABXA=AB
XB=BCXB=BCXB=BC
Correct answer: A
The original altitude lines are the perpendicular bisectors of the auxiliary sides. Their common point X is therefore the circumcenter of PQR, so its distances to the auxiliary vertices are equal.
This does not generally make X the circumcenter of the original triangle ABC.
A nondegenerate triangle has an orthocenter outside it. What can you conclude about its angles?
Correct answer: D
The acute case has an interior orthocenter and the right case has its orthocenter at a vertex. The outside case is therefore obtuse.
The orthocenter of nondegenerate triangle ABC is vertex A. What must the angle at A be?
The altitude line through B contains A, so line BA is perpendicular to AC. Therefore
A triangle has angles 50, 60, and 70 degrees. Where is its orthocenter?
Every supplied angle is less than 90 degrees. The triangle is acute, so its orthocenter lies strictly inside it.
In the auxiliary construction, PQ passes through C and is parallel to AB. Which original altitude line is its perpendicular bisector?
C is the midpoint of PQ. The altitude from C is perpendicular to AB, hence also to its parallel PQ.
Triangle ABC is obtuse at A. Which altitude feet lie beyond A on the side extensions?
The foot from B lies on AC extended past A, and the foot from C lies on AB extended past A. The foot from the obtuse vertex A lies inside BC.
AD is an altitude of triangle ABC. A point H, distinct from the vertices, lies on line AD and satisfies BH⊥ACBH\perp ACBH⊥AC. What follows?
H lies on the altitude lines from A and B. Their unique common point is the orthocenter, and the concurrence theorem puts it on the third altitude line as well.
AD is perpendicular to BC at D, but D is not its midpoint. What follows?
The right angle establishes the altitude. The missing midpoint condition prevents the same line from being the perpendicular bisector of BC.
Parallelograms ABCQ and ACBR arise in the auxiliary construction. Which equality between the two portions of QR follows?
Opposite sides give AQ the length BC, and AR the same length BC. Thus
In the equilateral proof, congruence makes angles ADB and ADC equal. Why are they right angles?
The two angles are supplementary because B, D, C are collinear. If their common measure is x, then
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