A triangle with one right angle is called a right triangle. The side opposite the right angle is called the hypotenuse of the triangle. The other two sides are calledlegs. The other two angles have no special name, but they are always complementary. Do you see why? The total angle sum of a triangle is 180 degrees, and the right angle is 90 degrees, so the other two must sum to 90 degrees.The triangle above has side c as its hypotenuse, sides a and b as its legs, and angle C as its right angle. Angles A andB are complementary.
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- Special Right Triangles
There are two types of right triangles that every mathematician should know very well. One is the right triangle formed when an altitude is drawn from a vertex of an equilateral triangle, forming two congruent right triangles. The angles of the triangle will be 30, 60, and 90 degrees, giving the triangle its name: 30-60-90 triangle. The ratio of side lengths in such triangles is always the same: if the leg opposite the 30 degree angle is of length x, the leg opposite the 60 degree angle will be of x, and the hypotenuse across from the right angle will be 2x. Here is a 30-60-90 triangle pictured below.
A special right triangle is a right triangle whose sides are in a particular ratio, called the Pythagorean Triples. You can also use the Pythagorean theorem ', but if you can see that it is a special triangle it can save you some calculations. The following figures show some examples of special right triangles and Pythagorean Triples. A 30-60-90 triangle is a special right triangle (a right triangle being any triangle that contains a 90 degree angle) that always has degree angles of 30 degrees, 60 degrees, and 90 degrees. Because it is a special triangle, it also has side length values which are always in a consistent relationship with one another.
The other common right triangle results from the pair of triangles created when a diagonal divides a square into two triangles. Each of these triangles is congruent, and has angles of measures 45, 45, and 90 degrees. If the legs opposite the 45 degree angles are of length x, the hypotenuse has a length of x. This ratio holds true for all 45-45-90 triangles. 45-45-90 triangles are also often called isosceles right triangles.
Solve for all pieces of the special right triangles. What is the measure of angle A in the triangle, rounded to the nearest degree? The degree measure of an angle in a right triangle is x, and sin x = 1 3. Which of these expressions are also equal to 1 3?
30 60 90 triangle rules and properties. The most important rule to remember is that this special right triangle has one right angle and its sides are in an easy-to-remember consistent relationship with one another - the ratio is a: a√3: 2a. The Math section of the SAT is full of special right triangles. The 5-12-13 triangle is a common triple. If you can recognize the triple pattern then you can easily calculate a missing side. Here is a list of the common triples associated with a 5 -12-13 right triangle.
One last characteristic to note is that the legs of a right triangle are also altitudes of the triangle. Therefore, the area of a right triangle is one-half the product of the lengths of its legs.
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.
Special Right Triangles Worksheet
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 Answer Key
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! |
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