12 multiple-choice questions, progressively harder.
Two right triangles have the same hypotenuse length c>0c>0c>0. One is 45-45-90 and the other is 30-60-90. Which listed side is longest?
Solution
Correct answer: D
Compare the positive lengths by their squares.
The first square is largest, so the 30-60-90 long leg is longest.
A square has diagonal 636\sqrt363 cm. An equilateral triangle has that diagonal length as its side length. What is the equilateral triangle's altitude?
Correct answer: B
The equilateral side is the stated diagonal length. Its altitude is half that side times 3\sqrt33.
Triangles ABCABCABC and BCDBCDBCD share side BCBCBC of length 666 cm. In ABCABCABC, angle CCC is right and angle AAA is 45∘45^\circ45∘. In BCDBCDBCD, angle BBB is right and angle CCC is 30∘30^\circ30∘. Find CD−ABCD-ABCD−AB.
Correct answer: C
In ABCABCABC, the shared side BCBCBC is a leg, so the hypotenuse is 626\sqrt262 cm. In BCDBCDBCD, BCBCBC is opposite 60∘60^\circ60∘, so it is the long leg.
An equilateral triangle has altitude 7157\sqrt{15}715 mm. What is its side length?
Correct answer: A
Divide the altitude by 3\sqrt33 to obtain half a side, then double.
A triangle has side lengths 555, 555, and 525\sqrt252 cm. Which conclusion is justified?
The longest side satisfies the Pythagorean equality.
The converse gives a right triangle. Its equal legs force equal acute angles, each 45∘45^\circ45∘.
A learner joins a vertex to the opposite midpoint in an equilateral triangle, then applies Pythagoras. Which statement supplies the missing justification?
The outer sides match, the base halves match, and the midpoint join is shared. SSS gives equal adjacent angles at the midpoint.
That proves the right-angle condition needed for Pythagoras.
In a 30-60-90 triangle, the long leg exceeds the short leg by 5(3−1)5(\sqrt3-1)5(3−1) cm. Find the short leg.
Let sss be the positive short leg. Subtract it from the long leg.
Cancellation is valid because 3−1\sqrt3-13−1 is positive.
A 45-45-90 triangle has hypotenuse 161616 cm. What is its perimeter?
Divide by 2\sqrt22 to recover each leg, then add both legs and the hypotenuse.
A square and an equilateral triangle have the same side length s>0s>0s>0. Let ddd be the square's diagonal and hhh the equilateral triangle's altitude. What is d/hd/hd/h?
The common positive side length cancels.
An equilateral triangle has side 434\sqrt343 cm. A square has diagonal equal to the triangle's altitude. What is the square's side length?
Find the altitude first, then treat it as the square's diagonal.
A square's diagonal and an equilateral triangle's side are both 121212 cm. Which comparison is correct?
The square side is 626\sqrt262 cm; the equilateral altitude is 636\sqrt363 cm. Compare their squares.
The equilateral altitude is longer because both lengths are positive.
A report says that a nondegenerate triangle has two 45∘45^\circ45∘ angles and side lengths in the ratio 1:3:21:\sqrt3:21:3:2. Can both statements hold?
Two 45∘45^\circ45∘ angles force equal opposite sides. The listed positive ratio gives three different side lengths.
Thus no such triangle satisfies both claims, regardless of rotation or scale.
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