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What's so special about the two right triangles shown here is that you have an even more special relationship between the measures of the sides — one that.
All 30-60-90-degree triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in radians, it translates to the following:
In any 30-60-90 triangle, you see the following:
- The shortest leg is across from the 30-degree angle.
- The length of the hypotenuse is always two times the length of the shortest leg.
- You can find the long leg by multiplying the short leg by the square root of 3.
Note: The hypotenuse is the longest side in a right triangle, which is different from the long leg. The long leg is the leg opposite the 60-degree angle.
The figure illustrates the ratio of the sides for the 30-60-90-degree triangle.
If you know one side of a 30-60-90 triangle, you can find the other two by using shortcuts. Here are the three situations you come across when doing these calculations:
- Type 1: You know the short leg (the side across from the 30-degree angle). Double its length to find the hypotenuse. You can multiply the short side by the square root of 3 to find the long leg.
- Type 2: You know the hypotenuse. Divide the hypotenuse by 2 to find the short side. Multiply this answer by the square root of 3 to find the long leg.
- Type 3: You know the long leg (the side across from the 60-degree angle). Divide this side by the square root of 3 to find the short side. Double that figure to find the hypotenuse.Finding the other sides of a 30-60-90 triangle when you know the hypotenuse.
In the triangle TRI in this figure, the hypotenuse is 14 inches long; how long are the other sides?
Because you have the hypotenuse TR = 14, you can divide by 2 to get the short side: RI = 7. Now you multiply this length by the square root of 3 to get the long side:
Recognizing special right triangles can provide a shortcut when answering some geometry questions. 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. Scroll down the page if you need more explanations about special right triangles, Pythagorean triples, videos and worksheets.
What is a 45°-45°-90° Triangle?
A 45°-45°-90° triangle is a special right triangle whose angles are 45°, 45° and 90°. The lengths of the sides of a 45°-45°-90° triangle are in the ratio of 1:1:√2.
A right triangle with two sides of equal lengths must be a 45°-45°-90° triangle.
You can also recognize a 45°-45°-90° triangle by the angles. A right triangle with a 45° angle must be a 45°-45°-90° special right triangle.
Side1 : Side2 : Hypotenuse = x:x:x√2
Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are both 3 inches.
Solution:
Step 1: This is a right triangle with two equal sides so it must be a 45°-45°-90° triangle.
Step 2: You are given that the both the sides are 3. If the first and second value of the ratio x:x:x√2 is 3 then the length of the third side is 3√2.
Answer: The length of the hypotenuse is 3√2 inches.
Example 2: Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 4√2 inches and one of the angles is 45°.
Solution:
Step 1: This is a right triangle with a 45°-45°-90° triangle.
You are given that the hypotenuse is 4√2. If the third value of the ratio n:n:n√2 is 4√2 then the lengths of the other two sides must 4.
Answer: The lengths of the two sides are both 4 inches. What is a 30°-60°-90° Triangle?
Another type of special right triangles is the 30°-60°-90° triangle. This is right triangle whose angles are 30°-60°-90°. The lengths of the sides of a 30°-60°-90° triangle are in the ratio of 1:√3:2.
You can also recognize a 30°-60°-90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30°-60°-90° special right triangle. A right triangle with a 30° angle or 60° angle must be a 30°-60°-90° special right triangle.
Side1 : Side2 : Hypotenuse = x:x√3:2x
Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and 4&dadic;3 inches.
Solution:
Step 1: Test the ratio of the lengths to see if it fits the n:n√2:2n ratio.
4:4√3:? = x:x√3:2xStep 2: Yes, it is a 30°-60°-90° triangle for x = 4Step 3: Calculate the third side. 2x = 2 × 4 = 8 Answer: The length of the hypotenuse is 8 inches.
Example 2: Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 8 inches and one of the angles is 30°.
Solution:
Step 1: This is a right triangle with a 30° angle so it must be a 30°-60°-90° triangle.
You are given that the hypotenuse is 8. Substituting 8 into the third value of the ratio x:x√3:2x, we get that 2x = 8 ⇒ x = 4.
Substituting x = 4 into the first and second value of the ratio we get that the other two sides are 4 and 4√3.
Answer: The lengths of the two sides are 4 inches and 4√3 inches. Special Triangles - Important Angles - 30°, 45°, 60°
45°-45°-90° Triangles, 30°-60°-90° Triangles.
- Show Step-by-step Solutions
The triangles are classified by side and by angle.
In this video you will learn:
1) 3-4-5 triangles and similar triangles
2) 5-12-13 triangles and similar triangles
3) 45-45-90 right triangle and similar triangles
4) 30-60-90 triangle and similar triangles
5) equilateral triangles
6) relationship between equilateral and 30-60-90 triangles. How to Solve Special Right Triangles?
When solving special right triangles, remember that a 30-60-90 triangle has a hypotenuse twice as long as one of the sides, and a 45-45-90 triangle has two equal sides.
- Show Step-by-step Solutions
45-45-90 and 30-60-90 degree triangles.
Discuss two special right triangles, how to derive the formulas to find the lengths of the sides of the triangles by knowing the length of one side, and a few examples using them.
What are Pythagorean Triples?
Any group of 3 integer values that satisfies the equation: a2 + b2 = c2 is called a Pythagorean Triple. Any triangle that has sides that form a Pythagorean Triple must be a right triangle. Some examples of Pythagorean Triple triangles are: 3-4-5 Triangles and 5-12-13 Triangles.
What is a 3-4-5 Triangle?
A 3-4-5 triangle is right triangle whose lengths are in the ratio of 3:4:5. When you are given the lengths of two sides of a right triangle, check the ratio of the lengths to see if it fits the 3:4:5 ratio.
Side1 : Side2 : Hypotenuse = 3n : 4n : 5n
Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 6 inches and 8 inches.
Solution:
Step 1:Test the ratio of the lengths to see if it fits the 3n : 4n : 5n ratio.
6 : 8 : ? = 3(2) : 4(2) : ? Step 2: Yes, it is a 3-4-5 triangle for n = 2.
Step 3: Calculate the third side
5n = 5 × 2 = 10 Answer: The length of the hypotenuse is 10 inches.
Example 2: Find the length of one side of a right triangle if the length of the hypotenuse is 15 inches and the length of the other side is 12 inches.
Solution:
Step 1: Test the ratio of the lengths to see if it fits the 3n : 4n : 5n ratio.
? : 12 : 15 = ? : 4(3) : 5(3) Step 2: Yes, it is a 3-4-5 triangle for n = 3.
Step 3: Calculate the third side
3n = 3 × 3 = 9 Answer: The length of the side is 9 inches.
What is a 5-12-13 Triangle?
What is a 5-12-13 Triangle?
A 5-12-13 triangle is a right-angled triangle whose lengths are in the ratio of 5:12:13. It is another example of a special right triangle.
Example:
3-4-5 and 5-12-13 are examples of the Pythagorean Triple. They are usually written as (3, 4, 5) and (5, 12, 13). In general, a Pythagorean triple consists of three positive integers such that a2 + b2 = c2. Two other commonly used Pythagorean Triples are (8, 15, 17) and (7, 24, 25)
Concepts and patterns of Pythagorean triplesExamples and families of Pythagorean Triples- Show Step-by-step Solutions
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