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Special Right Triangles quiz #1

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  • Which set of numbers can represent the side lengths of a 45-45-90 triangle?
    In a 45-45-90 triangle, the two legs are congruent and the hypotenuse is the leg length multiplied by √2. So, any set of numbers in the ratio x : x : x√2 can represent the side lengths.
  • In an isosceles triangle ABC, if the triangle is a 45-45-90 triangle, what is the measure of angle C?
    In a 45-45-90 triangle, the right angle is 90°, so angle C is 90°.
  • If the hypotenuse of a 45-45-90 triangle measures h units, what is the length of one leg of the triangle?
    The length of one leg is h / √2, which can be rationalized to (h√2) / 2.
  • How can you identify a 30-60-90 triangle based on its angle measures?
    A 30-60-90 triangle has angles measuring 30°, 60°, and 90°.
  • In an isosceles right triangle, what is the measure of one base angle?
    In an isosceles right triangle (45-45-90), each base angle measures 45°.
  • A triangle has sides with lengths of 9 meters, 9 meters, and 9√2 meters. What type of special triangle is this?
    This is a 45-45-90 triangle, since the legs are equal and the hypotenuse is the leg length times √2.
  • If the radius of a circle is r, what is the length of the side opposite the 30° angle in an inscribed 30-60-90 triangle?
    In a 30-60-90 triangle inscribed in a circle with radius r, the side opposite the 30° angle (the shortest leg) is r.
  • What is the measure of the central angle that corresponds to 1/6 of a circle?
    1/6 of a circle corresponds to a central angle of 60°.
  • If each leg of a 45-45-90 triangle measures 14 cm, what is the length of the hypotenuse?
    The hypotenuse is 14√2 cm.
  • What is an angle that measures 90 degrees called?
    An angle that measures 90 degrees is called a right angle.
  • What are the properties of a triangle that is both scalene and acute?
    A scalene acute triangle has all sides of different lengths and all angles less than 90°.
  • What is a scalene triangle?
    A scalene triangle is a triangle in which all three sides have different lengths.
  • What is the formula for the length of the altitude of an equilateral triangle with side length s?
    The altitude of an equilateral triangle with side length s is (s√3)/2.
  • What kind of triangle is always correct in its angle measures?
    An equilateral triangle is always correct in its angle measures, as all angles are 60°.
  • If the hypotenuse of a 45-45-90 triangle measures 18 cm, what is the length of one leg?
    The length of one leg is 18 / √2 = 9√2 cm (after rationalizing the denominator).
  • What fraction of a circle does a 45° central angle represent?
    A 45° central angle represents 1/8 of a circle.
  • How many sides does a regular polygon have if each interior angle measures 60°?
    A regular polygon with each interior angle measuring 60° is an equilateral triangle, which has 3 sides.
  • What are the possible side length ratios for a 30-60-90 triangle?
    The side lengths of a 30-60-90 triangle are in the ratio 1 : √3 : 2, corresponding to the sides opposite the 30°, 60°, and 90° angles, respectively.
  • What is the multiplier used to find the hypotenuse of a 45-45-90 triangle when given the length of one leg?
    In a 45-45-90 triangle, the hypotenuse is found by multiplying the length of one leg by the square root of 2. That is, hypotenuse = leg × √2.
  • How do you find the length of a leg in a 45-45-90 triangle if you are given the hypotenuse?
    To find the length of a leg in a 45-45-90 triangle when given the hypotenuse, divide the hypotenuse by the square root of 2 and rationalize if needed: leg = hypotenuse ÷ √2 = (hypotenuse × √2) / 2.