Special Right Triangles

Key Questions

• What are the basic properties of a 45-45-90 triangle?

  Answer:

  Consider the properties of the sides, the angles and the symmetry.

  Explanation:

  `45-45-90" "` refers to the angles of the triangle.

  The `color(blue)("sum of the angles is " 180°)`

  There are `color(blue)("two equal angles")`, so this is an isosceles triangle.

  It therefore also has `color(blue)(" two equal sides.")`

  The third angle is `90°`. It is a `color(blue)("right-angled triangle")` therefore Pythagoras' Theorem can be used.

  The `color(blue)("sides are in the ratio " 1 :1: sqrt2)`

  It has `color(blue)("one line of symmetry")` - the perpendicular bisector of the base (the hypotenuse) passes through the vertex, (the `90°` angle).

  It has `color(blue)("no rotational symmetry.")`

  EZ as pi · 1 · Apr 21 2018
• What are the basic properties of a 30-60-90 triangle?

  `mathbf{30^circ"-"60^circ"-"90^circ}` Triangle

  The ratios of three sides of a `30^circ"-"60^circ"-"90^circ` triangle are:

  `1:sqrt{3}:2`

  ---

  I hope that this was helpful.

  Wataru · 1 · Nov 6 2014
• How do the Special Right Triangles relate to the Unit Circle?

  Each black-and-red (or black-and-yellow) triangles is a special right-angled triangle. The figures outside the circle - `pi/6, pi/4, pi/3` - are the angles that the triangles make with the horizontal (x) axis. The other figures - `1/2, sqrt(2)/2, sqrt(3)/2` - are the distances along the axes - and the answers to `sin(x)` (yellow) and `cos(x)` (red) for each angle.

  CJ · 5 · Dec 13 2014
• What are the Special Right Triangles?

  Special Right Triangles

  1. `30^circ`-`60^circ`-`90^circ` Triangles whose sides have the ratio `1:sqrt{3}:2`

  2. `45^circ`-`45^circ`-`90^circ` Triangles whose sides have the ratio `1:1:sqrt{2}`

  ---

  These are useful since they allow us to find the values of trigonometric functions of multiples of `30^circ` and `45^circ`.

  Wataru · 1 · Dec 7 2014

• What are the Special Right Triangles?
• What are the basic properties of a 45-45-90 triangle?
• What are the basic properties of a 30-60-90 triangle?
• How do the Special Right Triangles relate to the Unit Circle?
• How do you find the other two side of a right triangle ABC, if `∠B=60` and `AB=12` where AB is the hypotenuse?
• How do you use special right triangles to find the missing side lengths?
• Why do you need to use special right triangles?
• How do you know when to use the special right triangles?
• What is the length of the shorter diagonal of the parallelogram if the lengths of the two adjacent sides of parallelogram ABCD are 8 and 14 units respectively and if the degree measure of the included angle is 60º?
• What is the length of the hypotenuse, in a 30
  -60
  -90
  triangle the length of the shorter leg is 8 units?
• In a 45-45-90 triangle the hypotenuse is 7, what are the lengths of the other legs?
• What is the reference angle and what quadrant does 3.3 lie?
• What is the reference angle and what quadrant does 300 degrees lie?
• How do you solve the triangle if the angles given are 90, 60, and 30 and the length of the hypotenuse is given as 10?
• In triangle 45, 45, 90 DEF triangle, angle D is congruent to angle F. The perimeter of triangle DEF is 35 mm, and DF = 14mm, how do you determine DE?
• How can I find length of side of equilateral triangle given only altitude which is `12sqrt3`?
• How do you find length of sides of an equilateral triangle inscribed in a circle with a radius of 36?
• What is the side of regular polygon of 6 sides inscribed in a circle with a radius of 15 in?
• There is a 30, 60, 90 triangle with a short side of 5 and the hypotenuse is 10, how do I find the long leg?
• How do you find the missing length of an isosceles triangle given Base = 8 Angle = 30?
• How do you find the missing length of an isosceles triangle given Base = 20 Angle = 25?
• How do you solve the right triangle given the hypotenuse is 12 and it is a 30-60-90 triangle?
• How do you solve the right triangle given the hypotenuse is 4 and it is a 45-45-90 triangle?
• How do you find trigonometric ratios of 30, 45, and 60 degrees?
• How do you find the area of a 30, 60, 90, triangle?
• If you have a right triangle 30,60,90 degree angles and you have the smallest size (width) to be 4 how do I solve for x (the length of the triangle)?
• In a 45° - 45° - 90° right triangle, the length of the hypotenuse is `15sqrt2`, what is the length on one of the legs?
• Are 20, 48, and 52 the side measurements of a special right triangle?
• The shorter leg of a 30°- 60°- 90° right triangle is 7.5 inches. How long is the longer leg and the hypotenuse?
• The hypotenuse of a 30-60-90 right triangle measure 18cm. What are the lengths of the longer leg and shorter leg?
• Andrew claims that a wooden bookend in the shape of a 45°- 45°- 90° right triangle has side lengths of 5 in., 5 in., and 8 in. Is he correct? If so, show the work and if not, show why not.
• In a 45°- 45°- 90° right triangle, the length of the hypotenuse is 12. How long are the legs?
• How do you find the exact values `sin (pi/4)` using the special triangles?
• How do you find the exact values `sin (pi/6)` using the special triangles?
• How do you find the exact values `cos (pi/4)` using the special triangles?
• How do you find the exact values `cos (pi/6)` using the special triangles?
• Carrie found a stone in the shape of a 30°–60°–90° triangle. The hypotenuse has a length of 5 cm. What is the length of the shorter leg?
• How do you use special angles to find the ratios of csc 315 degrees?
• How do you use special angles to find the ratios of sec 180 degrees?
• How do you use special angles to find the ratios of sin 150 degrees?
• The Hyptoneus is 16, inside/above to the left of the 90 degrees angle is 56 degrees, how do you find the missing angle?
• How do you know if to divide or multiply in soh cah toa problem for example Sin30=x/5?
• How do you find the unknown angle if the adjacent of the triangle is 9 cm , the hypotenuse is 14 cm and the opposite side is unknown?
• If you have a side that is 4cm and then an angle that is 25degrees what formula do you use to figure out all the sides?
• There are 3 angles to the triangle but how do you know which side uses Soh, Cah, and Toa to be found?
• If the triangle has an angle of 30 degrees, what is the Hypotonese And The Opposite?
• A triangle with angle A 15 degrees, and sides AB=AC=10cm, how do you find the all the angles and sides of the triangle?
• What are the coordinates of -2 pi radians on the unit circle?
• Where is -10pi/ 3 on the Unit Circle?
• How do you create a 16 point unit circle that ranges from 0 to 8pi?
• How do you find the point on the unit circle that corresponds to pi/4?
• How do you find the point on the unit circle that corresponds to 5pi/4?
• How do you find the length of the legs of a 45°-45°-90° triangle with a hypotenuse of 4* sqrt 2 cm.?
• The shorter leg of a 30°- 60°- 90° right triangle is 12.5 inches. How long are the longer leg and the hypotenuse?
• The hypotenuse of a 30°- 60°- 90° right triangle measures 22 cm. What are the lengths of the longer leg and shorter leg?
• Andrew claims that a wooden bookend in the shape of a 45°- 45°- 90° right triangle has side lengths of 9 in., 9 in., and 11 in. Is he correct?
• In a 45°- 45°- 90° right triangle, the length of the hypotenuse is 15. How long are the legs?
• In a 45 degree-45 degree right triangle, the length of a hypotenuse is 22 ft. How long is a leg of the triangle?
• In a right triangle, one with angle measures of 30-60-90. The hypotenuse is y, the shortest side is x, and the last side is 21, what are the lengths of x and y?
• Each side of a rhombus is 14 in. Long. Two of the sides form a 60 degree angle, how do you find the area of the rhombus?
• The hypotenuse of an isosceles right triangle is 8.4 in. How do you find the length of a side to the nearest tenth of an inch?
• How do you solve the right triangle given the Hypotenuse side is 7 and it is a 45-45-90 triangle?
• Is a triangle with sides of 6, 8, and 10, a right triangle?
• Is a triangle with sides of 3,4,6 a right triangle?
• Is a triangle with sides of 3/4, 1, 1/5 a right triangle?
• How and when do we use SOHCAHTOA?
• In right triangle ABC, angle C is the right angle and angle B=30. What's the ratio of AC to BC?
• Question #d9863
• How do you find the geometric mean between 5 and 20?
• How do you find the geometric mean between 36 and 4?
• How do you find the geometric mean between 40 and 15?
• How do you find the geometric mean between 20 and 25?
• How do you find the geometric mean between 36 and 24?
• How do you find the geometric mean between 12 and 2.4?
• How do you find the geometric mean between 18 and 1.5?
• What is `tan(45)`, `sin(45)` and `cos(45)`?
• How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the shortest side is 1?
• How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the longest side is 8?
• How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the longest side is 5?
• How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the side opposite `60^circ` is 6?
• How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the side opposite `60^circ` is 4?
• An escalator in a department store is to carry people a vertical distance of 20 feet between floors. How long is the escalator if it makes an angle of `30^circ` with the ground?
• An escalator in a department store is to carry people a vertical distance of 20 feet between floors. How long is the escalator if it makes an angle of `60^circ` with the ground?
• How do you find the remaining sides of a `45^circ-45^circ-90^circ` triangle if the shortest sides are each 4/5?
• How do you find the remaining sides of a `45^circ-45^circ-90^circ` triangle if the shortest sides are each 1/2?
• How do you find the remaining sides of a `45^circ-45^circ-90^circ` triangle if the longest sides are each `8sqrt2`?
• How do you find the remaining sides of a `45^circ-45^circ-90^circ` triangle if the longest sides are each `5sqrt2`?
• How do you find the remaining sides of a `45^circ-45^circ-90^circ` triangle if the longest side is 12?
• A Bullet is fired into the air at an angle of `45^circ`. How far does it travel before it is 1,000 feet above the ground?
• Question #0c69f