Recognizing special right
triangles in geometry can help you to answer some questions quicker. A special
right triangle is a right triangle whose sides are in a particular
ratio. You can also use the Pythagorean
theorem, but if you can see that it is a special triangle it can
save you some calculations.
There are three types of special triangles:
3-4-5 triangles, 45°-45°-90°
triangles and 30°-60°-90° triangles. In this lesson we will study 3-4-5 triangles.
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.
The following video gives another example of the 3-4-5 triangle