12 multiple-choice questions, progressively harder.
Which description defines an altitude of a nondegenerate triangle?
Solution
Correct answer: C
An altitude has one endpoint at a vertex and is perpendicular to the line of the opposite side. That line may have to be extended.
Can the foot of an altitude lie outside its opposite side segment?
Correct answer: A
The definition uses the full line containing the opposite side. In an obtuse triangle, two perpendicular feet lie on extensions.
What is the orthocenter of a nondegenerate triangle?
Correct answer: D
The altitude lines are concurrent, and their common point is the orthocenter. The definition does not require the point to lie inside the triangle.
Where is the orthocenter of an acute triangle?
Correct answer: B
In an acute triangle the altitude feet are inside the sides. Two altitude segments cross inside the triangle, locating the orthocenter.
Triangle ABC is right at C. Where is its orthocenter?
The legs AC and BC are already altitudes because they are perpendicular. They meet at C, which is also on the third altitude.
Where is the orthocenter of an obtuse triangle?
Two altitude lines run outside the triangular interior and intersect at an exterior point. The foot from an acute vertex lies beyond the obtuse vertex.
To construct the altitude from B in triangle ABC, which line must receive the perpendicular from B?
The side opposite vertex B is AC. Construct the perpendicular from B to its complete supporting line, extending AC when needed.
Which given mark directly establishes that AD is an altitude from A?
The square at D supplies the perpendicular condition. Equal side halves or equal vertex-angle parts would describe other special segments.
The auxiliary side QR passes through A and is parallel to BC. Why is the altitude line from A perpendicular to QR?
The altitude line is perpendicular to BC by definition. Since QR is parallel to BC, parallel-line angle relationships transfer the right angle to QR.
In equilateral triangle ABC, D is the midpoint of BC and the whole median AD is 24 cm. H is the orthocenter. How far is H from D?
In an equilateral triangle, the orthocenter and centroid are the same point. The centroid-to-side-midpoint portion is one third of the whole median.
In the auxiliary-triangle proof, which fact about A is needed in addition to perpendicularity to QR?
A perpendicular bisector must pass through the midpoint of the segment it bisects. The required midpoint belongs to auxiliary side QR.
Which center is defined by equal distances from the three vertices?
The perpendicular-bisector theorem makes the circumcenter equally far from the vertices. Orthocenter means the intersection of altitude lines, which is a different condition.
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