Geometry Practice Test: Triangle Inequality, Midsegments and Triangle Centers
- Triangle Inequality, Midsegments and Centers
- 10 questions
- 13 minutes
These 10 questions cover which side lengths can form a triangle, the Triangle Midsegment Theorem and the special points where segments in a triangle meet. Tip: the two shorter sides must add up to more than the longest side.
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Question 1
Which set of numbers could be the side lengths of a triangle?
Correct answer: C
The two shorter sides must sum to more than the longest: 5 + 7 = 12 > 11. For 2, 6, 8 the sum is exactly 8, which only gives a flat segment, not a triangle.
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Question 2
Two sides of a triangle are 6 and 10. Which describes all possible lengths x of the third side?
Correct answer: B
The third side must be greater than the difference 10 − 6 = 4 and less than the sum 10 + 6 = 16. The endpoints themselves would give a flat figure.
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Question 3
In triangle ABC, AB = 8, BC = 12 and AC = 15. Which angle is the largest?
Correct answer: D
The largest angle is opposite the longest side. AC = 15 is the longest side, and the angle opposite it is ∠B.
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Question 4
A midsegment of a triangle is parallel to a side that is 18 cm long. How long is the midsegment?
Correct answer: A
By the Triangle Midsegment Theorem, a midsegment is parallel to the third side and half as long: 18 ÷ 2 = 9 cm.
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Question 5
In triangle ABC, DE is the midsegment parallel to BC. DE = 3x − 1 and BC = 4x + 8. What is BC?
Correct answer: C
BC = 2 · DE, so 4x + 8 = 2(3x − 1) = 6x − 2. Then x = 5 and BC = 4(5) + 8 = 28. The value 14 is the midsegment DE.
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Question 6
A triangle has a perimeter of 40 cm. Its three midsegments form a smaller triangle. What is the perimeter of the smaller triangle?
Correct answer: B
Each midsegment is half as long as the side it is parallel to, so the midsegment triangle has half the perimeter: 20 cm.
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Question 7
In triangle ABC, AM is a median of length 18, and G is the centroid. What is AG?
Correct answer: D
The centroid divides each median in the ratio 2 : 1 from the vertex, so AG = 23 · 18 = 12 and GM = 6.
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Question 8
Which point of a triangle is the same distance from all three vertices?
Correct answer: A
Every point on the perpendicular bisector of a side is equidistant from that side's endpoints, so the circumcenter is equidistant from all three vertices. The incenter is equidistant from the sides instead.
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Question 9
Three straight walking paths form a triangle around a park. Planners want a fountain inside the triangle that is the same distance from all three paths. Where should it go?
Correct answer: B
The incenter, where the angle bisectors meet, is equidistant from the three sides of the triangle. The circumcenter is equidistant from the corners, not the sides.
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Question 10
Where is the circumcenter of a right triangle?
Correct answer: C
The hypotenuse of a right triangle is a diameter of its circumscribed circle, so the circumcenter is the midpoint of the hypotenuse.
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