As we know, a right triangle is a triangle whose one angle always measures 90°. The longest side opposite of a right triangle is called the hypotenuse, while the horizontal leg is called the base and the vertical leg is called the height or the altitude.
Special Right Triangles Date Period Find the missing side lengths. Leave your answers as radicals in simplest form. 1) a 2 2 b 45° 2) 4 x y 45° 3) x y 3 2 2 45° 4) x y 3 2 45° 5) 6 x y 45° 6) 2 6 y x 45° 7) 16 x y 60° 8) u v 2 30°-1. There are two 'special' right triangles that will continually appear throughout your study of mathematics: the 30º-60º-90º triangle and the 45º-45º-90º triangle. The special nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions. Visit www.doucehouse.com for more videos like this. In this video, I explain the basics behind the 45-45-90 and 30-60-90 special right triangles.
What is a Special Right Triangle
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an 'angle-based' right triangle. Every right triangle has the property that the sum of the squares of the two legs is equal to the square of the hypotenuse (the longest side). The Pythagorean theorem is written: a2 + b2 = c2. What’s so special about the two right triangles shown here is that you have an even more special.
Special right triangles are right triangles whose angles or sides are in a particular ratio. They have some regular features that make calculations on it much easier.
In geometry, the Pythagorean Theorem is commonly used to find the relationship between the sides of a right triangle, given by the equation: a2 + b2 = c2, where a, b denotes the height and base of the triangle and c is the hypotenuse. But, if it is a special right triangle, we can use simpler formulas.
Some common special right triangles are shown below.
Types of Special Right Triangle
The two most common special right triangles are:
45-45-90 Triangle
A 45-45-90 triangle is a special right triangle whose three angles measure 45°, 45° and 90°. The ratio of its side lengths (base: height: hypotenuse) is 1: 1: √2.
30-60-90 Triangle
A 30-60-90triangle is a special right triangle whose three angles measure 30°, 60° and 90°. The ratio of its side lengths (base: height: hypotenuse) is1: √3: 2.
Apart from the above two types, there are some other special right triangles.
Others: Pythagorean Triples
Some right triangles have sides that are of integer lengths and are collectively called the Pythagorean triples. Such triangles can be easily remembered and any multiple of the sides produces the same relationship. Pythagorean triples can be of three types:
- Common Pythagorean triples: Sides with integer lengths. Examples – 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40- 41.
- Almost-isosceles Pythagorean Triples: Sides with integer lengths but almost isosceles. Examples – 20-21-29, 119- 120-169, 696-697-985, and 4,059-4,060-5,741.
- Sides that are in Geometric Progression: Also known as the Kepler triangle, if the sides are in geometric progression a, ar, ar2, its common ratio r is given by r = √φ where φ is the golden ratio.
Formulas
Although there is no common formula for special right triangles, each of them has specific formulas for finding the missing sides, area, and perimeter based on the ratio of their side lengths. Find their formulas with solved examples in our separate articles.
How to Solve Special Right Triangles
Solving special right triangles is about finding the missing lengths of the sides. Instead of using the Pythagorean Theorem, we can simply use the special right triangle ratios to find the missing length. Let us understand the concept better by doing some practice problems.
The hypotenuse of a 45-45-90 triangle is 12√2 mm. Calculate the length of its base and height.
As we know,
Ratio of their Side Lengths = x: x: x√2, here x√2 = hypotenuse = 12√2 mm
Thus,
x√2 = 12√2 mm
Squaring both sides we get,
⇒ (x√2)2 = (12√2)2 mm
⇒ 2x2 = 144 x 2 = 288
⇒ x = √144 = 12 mm
Hence, the base and height of the given right triangle measure 16.97 mm each.
The longer side of a 30-60-90 right triangle is given by 5√3 cm. What is the measure of its shorter side and hypotenuse?
As we know,
Ratio of their Side Lengths = x: x√3: 2x, here x = shorter side, x√3 = longer side = 5√3 cm, 2x = hypotenuse
Thus,
x√3 = 5√3
Squaring both sides we get,
(x√3)2 = (5√3)2
⇒ 3x2 = 25 x 3
⇒ x2 = 25
⇒ x = 5 cm
The length of the hypotenuse and the other side of a right triangle are 30 cm and 24 cm, respectively. Find the length of the missing side.
As we know,
Here we have to find whether the sides are in the ratio of 3x: 4x: 5x
Thus,
?: 24: 30 = ?: 4(6): 5(6)
Thus, the sides are in the ratio of 3x: 4x: 5x and it is a 3-4-5 triangle.
For calculating the third side,
For, n = 6
Hence, the length of the other side,
3x = 3 x 6 = 18
Therefore the length of the missing side is 18 cm.
If the two sides of a right triangle are 3 ft and 4 ft, find the length of hypotenuse.
As we know,
Here we have to find whether the sides are in the ratio of 3x: 4x: 5x
3: 4: ? = 3(1): 4(1): ?
Thus, the sides are in the ratio of 3x: 4x: 5x and it is a 3-4-5 triangle
For calculating the hypotenuse,
For, n = 1
Hence, the length of the hypotenuse,
5x = 5 x 1 = 5
Therefore the length of the hypotenuse is 5 ft
Special Right Triangle: 45º-45º-90º Isosceles Right Triangle MathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use Contact Person:Donna Roberts |
There are two 'special' right triangles that will continually appear throughout your study of mathematics: the 30º-60º-90º triangle and the 45º-45º-90º triangle. The special nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions. This page will deal with the 45º-45º-90º triangle.
All 45º-45º-90º triangles are similar! They satisfy Angle -Angle (AA) for proving trianlges similar. |
Our first observation is that a 45º-45º-90º triangle is an 'isosceles right triangle'. This tells us that if we know the length of one of the legs, we will know the length of the other leg. This will reduce our work when trying to find the sides of the triangle. Remember that an isosceles triangle has two congruent sides and congruent base angles (in this case 45º and 45º).
Congruent 45º-45º-90º triangles are formed when a diagonal is drawn in a square. Remember that a square contains 4 right angles and its diagonal bisects the angles. If the side of the square is set to a length of 1 unit, the Pythagorean Theorem will find the length of the diagonal to be units. | |
Note: the side of the square need not be a length of 1 for the patterns to emerge. The choice of a side length of 1 simply makes the calculations easier.
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Once the sides of the 45º-45º-90º triangle are established, a series of relationships (patterns) can be identified between the sides of the triangle. ALL 45º-45º-90º triangles will possess these same patterns. These relationships will be referred to as 'short cut formulas' that can quickly answer questions regarding side lengths of 45º-45º-90º triangles, without having to apply any other strategies such as the Pythagorean Theorem or trigonometric functions. |
Since 45º-45º-90º triangles are similar, their corresponding sides are proportional. As such, we can establish a pattern as to how their sides are related. The following pattern formulas will let you quickly find the sides of a 45º-45º-90º triangle even when you are given only ONE side of the triangle. Remember, these formulas work ONLY in a 45º-45º-90º triangle!
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to get the formula where the answer is already rationalized. You can use either formula to find the leg.
This example shows the application of the patterns when the leg is given.
Always look at what is 'given' and what you need to find.
Find x and y. | x is the 'other' leg (isosceles → legs equal) No formula needed. x = 9Answer | y is the hypotenuse (across from the 90º angle)
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This example shows the application of the patterns when the hypotenuse is given.
Always start with what is 'given' and work from that point.
Special Right Triangles 30 60 90
Find x and y. | x and y are the legs (12 is the hypotenuse) x = ½ • 12 • x = 6 Answer | y is the 'other' leg (use the value for x) No formula needed. y = 6 Answer |
where you need to rationalize the final answer.
This example requires more work with radicals. For a review on radicals, see Radical Review.
Find x and y. | 8is the leg (x is the 'other' leg) No formula needed. |
Notice that when you are working with a 45º-45º-90º triangle you are working with. Think of the TWO being related to the FOUR: 45, 45, When you work with 30º-60º-90º and 45º-45º-90º triangles, you will need to keep straight which radical goes with which triangle. |
I forgot the formula patterns! Now what? When working with a 45º-45º-90º triangle, you can always use the Pythagorean Theorem. Unlike the 30º-60º-90º triangle, in a 45º-45º-90º triangle you always know, or can represent, two sides of the triangle. • If you know the length of a leg, you know both legs. • If you know the length of the hypotenuse, represent the legs as x and x. The Pythagorean Theorem will always work! |
Special Right Triangles Quizlet
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