Math by First Principles
Level 2 · Intermediate ← Back to lesson

Special Right Triangles: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered
Question 1 of 12
  1. 1

    In triangle ABCABC, angle CC is right and angle AA is 30∘30^\circ. Side ACAC measures 535\sqrt3 cm. Find ABAB.

    Triangle relationshipsSchematic for interpreting stated givens, not for measuring. Right triangle CAB has its marked right angle at C. The angle at A is 30 degrees and the angle at B is 60 degrees. Side CA is labeled 5√3 cm. Hypotenuse AB is labeled AB.5√3 cmABCA30°B60°
  2. 2

    The perimeter of a 45-45-90 triangle is (14+72)(14+7\sqrt2) cm. What is either leg?

  3. 3

    A square has diagonal 10510\sqrt5 mm. Find the exact side length.

    Triangle relationshipsSchematic for interpreting stated givens, not for measuring. Square PQRS has all four equal sides and four right angles. Diagonal PR is dashed. Side PQ is labeled s. Diagonal PR is labeled 10√5 mm.PQRSs10√5 mm
  4. 4

    An equilateral triangle has altitude 737\sqrt3 cm. What is one side length?

    Triangle relationshipsSchematic for interpreting stated givens, not for measuring. Equilateral triangle ABC has midpoint M on BC and segment AM. Matching single ticks mark the equal outer sides AB and AC. The proved right angle at M and equal half-base double ticks are shown. The sloping sides are labeled e. AM is labeled 7√3 cm.ABCMeeAM: 7√3 cm30°30°
  5. 5

    A 30-60-90 triangle has long leg 464\sqrt6 m. What is its short leg?

    Triangle relationshipsSchematic for interpreting stated givens, not for measuring. Right triangle CAB has its marked right angle at C. The angle at A is 30 degrees and the angle at B is 60 degrees. Side CA is labeled 4√6 m. Side CB is labeled s. 4√6 msCA30°B60°
  6. 6

    An equal-leg right triangle has hypotenuse 131013\sqrt{10} units. What is the sum of its two legs?

    Triangle relationshipsSchematic for interpreting stated givens, not for measuring. Right triangle CAB has its marked right angle at C. The two legs have matching ticks. Side CA is labeled s. Side CB is labeled s. Hypotenuse AB is labeled 13√10.ss13√10CA45°B45°
  7. 7

    An equilateral triangle has side length 626\sqrt2 cm. Find its altitude.

    Triangle relationshipsSchematic for interpreting stated givens, not for measuring. Equilateral triangle ABC has midpoint M on BC and segment AM. Matching single ticks mark the equal outer sides AB and AC. The proved right angle at M and equal half-base double ticks are shown. The sloping sides are labeled 6√2 cm. AM is labeled h.ABCM6√2 cm6√2 cmAM: h30°30°
  8. 8

    In a 30-60-90 triangle, the hypotenuse is 88 cm longer than the short leg. What is the long leg?

  9. 9

    In equilateral triangle ABCABC, MM is the midpoint of BCBC. Before assuming a right angle, which evidence proves triangles ABMABM and ACMACM congruent?

    Triangle relationshipsSchematic for interpreting stated givens, not for measuring. Equilateral triangle ABC has midpoint M on BC and segment AM. Matching single ticks mark the equal outer sides AB and AC. No perpendicular or angle-bisection conclusion is marked. ABCM
  10. 10

    A 45-45-90 triangle has a leg of (3x+2)(3x+2) cm and hypotenuse (8x−1)2(8x-1)\sqrt2 cm. Both lengths are positive. Find xx.

  11. 11

    The perimeter of a 30-60-90 triangle is (18+63)(18+6\sqrt3) m. What is its short leg?

  12. 12

    A triangle has side lengths 7\sqrt7, 21\sqrt{21}, and 272\sqrt7 units. What is the angle opposite the side of length 21\sqrt{21}?

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