12 multiple-choice questions, progressively harder.
Positive lengths 888, 151515, and zzz are proposed, with z≥15z\ge15z≥15. Which condition gives an obtuse nondegenerate triangle?
Solution
Correct answer: D
Existence requires
For an obtuse triangle, the longest side must also satisfy
Combining these strict bounds gives the stated open interval.
A triangle has sides 999, 121212, and 161616 units. Which claim and reason are correct?
Correct answer: B
The lengths close.
Their squared comparison is
The larger longest-side square gives an obtuse triangle.
Which set forms an acute nondegenerate triangle?
Correct answer: C
For the selected set, 5 plus 7 exceeds 8, so it exists.
The longest-side square is smaller. The 4,5,8 triangle is obtuse, 2,3,5 is flat, and 7,24,25 is right.
To prove the converse, construct a right triangle with legs equal to two sides of the given triangle. After proving the third sides equal, which congruence criterion transfers the right angle?
Correct answer: A
Call the given side lengths a, b, and c, with c the side opposite the angle being tested. The constructed triangle has legs a and b and hypotenuse d. Applying the theorem only to it gives
The converse hypothesis makes the right side equal to the square of c. Positivity gives
All three corresponding side lengths now match, which is SSS. The original triangle was not known to be right; its right angle follows after congruence.
A right triangle has legs xxx and x+7x+7x+7 cm and hypotenuse x+8x+8x+8 cm, with x>0x>0x>0. Find xxx.
Apply the theorem using the stated hypotenuse.
The roots are 5 and negative 3. Positivity leaves x equal to 5; the lengths are 5, 12, and 13 cm.
Two right triangles have equal hypotenuse lengths. The first has legs 999 and 101010 units. The second has one leg 111111 units. Find the second triangle's other leg.
The first hypotenuse has square
That square is 181. Subtract the second triangle's known leg square.
An isosceles triangle has side lengths 101010, 101010, and 121212 units. What is its angle classification?
The lengths close because 10 plus 10 exceeds 12. The 12-unit side is longest.
The largest angle is acute, so all three angles are acute.
The similarity proof uses
Here p names AH, q names HB, and c names AB. Which incidence fact justifies the equality?
Segment addition applies because the two subsegments partition AB.
No midpoint or equal-leg hypothesis is required.
A triangle has side lengths 161616, 202020, and an integer xxx, with 0<x≤200<x\le200<x≤20. What is the smallest value of xxx making an acute nondegenerate triangle?
Since 20 is longest, existence requires x greater than 4. Acute classification further requires
The smallest admissible integer is 13, and it also satisfies the closure and upper bounds.
Which calculation shows that adding 222 to every length of the triple (3,4,5)(3,4,5)(3,4,5) does not preserve a right triangle?
Adding 2 produces the valid side set 5, 6, and 7. Compare
These are unequal. The other options check the original triple or its multiples, not a common addition.
A right triangle has legs exactly 0.90.90.9 m and 120120120 cm. Find its hypotenuse in centimeters.
Convert to a common unit first: 0.9 m is 90 cm. Then
The hypotenuse is 150 cm.
A right triangle has legs a,ba,ba,b and hypotenuse ccc. For every positive scale factor kkk, what is the hypotenuse after all sides are scaled by kkk?
Every side length is multiplied by k. The squared equality verifies the same result:
The positive hypotenuse is kc.
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