12 multiple-choice questions, progressively harder.
Two sides measure 4 and 9 units. Which proposed third-side length gives a nondegenerate triangle?
Solution
Correct answer: C
The whole range is
Only 10 is strictly inside it. The endpoint lengths 5 and 13 are flat, while 14 is too long.
Two triangle sides measure 8 and 10 units. How many integer lengths are possible for the third side?
Correct answer: A
The strict bounds are
The integers run from 3 through 17, inclusive. Their number is
Do not count the two flat boundaries.
Proposed triangle sides are t plus 2, t plus 5, and 13 units, where t is real. Which values of t give a nondegenerate triangle?
Correct answer: D
The first pair comparison gives
Thus
The other two comparisons reduce to the true numerical statements 15 greater than 5 and 18 greater than 2. When t exceeds 3, all side lengths are positive.
The proposed side lengths are 2u minus 1, 9, and 12 units. What is the complete range of real u for a nondegenerate triangle?
Correct answer: B
The side expression must lie between the difference and sum of 9 and 12:
Add 1 to each part and divide by 2:
This also makes every side positive.
The same positive length x is a third side for one triangle with fixed sides 5 and 9, and another with fixed sides 7 and 12. What is the complete range of x?
The first triangle requires
The second requires
Keep only values in both intervals:
The stronger lower bound is 5 and the stronger upper bound is 14.
A triangle has perimeter 29 units. One side is 11 units; another is x units, so the third is 18 minus x units. What is the complete range of x?
Comparing x and the 11-unit side with the third gives
Hence
The other nontrivial comparison is
The remaining pair sums to 18, already greater than 11. Both variable sides are positive throughout the stated interval.
Triangle ABC has AB of length 9 and AC of length 4. Extend BA beyond A to D so AD equals AC. Which conclusion follows for BC?
The extension has total length
The extension proof gives BD strictly longer than BC, so
BC<5BC<5BC<5 is impossible because the triangle inequality also requires
A, B, and P lie on a line in that order. AP measures 15 meters, and BP measures 4 meters. How much longer is the route A to P to B than the direct segment AB?
Since B lies between A and P,
The direct distance is 11 meters. The route goes past B and returns:
The route length is 19 meters, so its extra length is
The 4-meter overshoot is traveled twice.
A route goes from A through B and C to D, with consecutive leg lengths 5, 7, and 9 units. A, B, and C are noncollinear. Which bound is certain?
The first bend makes
The final leg gives
Combining those steps yields
The final comparison stays strict because the first replacement already shortened the route.
A nondegenerate quadrilateral has one side of length 18 units. Its other sides have lengths x, 7, and 6 units. Which restriction is necessary when you compare the 18-unit side with the other three together?
The polygon inequality requires
Subtracting 13 gives
Equality would make that route flat. This comparison alone is not a full set of conditions on the quadrilateral.
A triangle is proposed with side lengths k, 2k, and 10 units, where k is real. Which is its complete valid range?
The comparison of the two variable sides with 10 gives
The comparison with the 2k side gives
Thus the necessary bounds are
They make k positive. The third inequality, 2k plus 10 greater than k, then holds too.
Two fixed sides measure 10 and 16 units. A proposed rule for the third side x says 6≤x<266\le x<266≤x<26. What is the corrected rule?
At a third-side length of 6, the shorter lengths give
This is flat. Both the difference and sum boundaries must be strict:
The lower endpoint cannot be included.
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