I don¡¯t know how to slove this problem can someone help me£¿
The shorter leg of a 30¡ã60¡ã90¡ãtriangle is 2.5 feet long.Find the perimeter.
the sides are in the ratio 1:√3:2
multiply all those by 2.5 and you have a perimeter of (5/2)(3+√3)
Thank you£¡
To solve this problem, we need to find the lengths of the other two sides of the triangle and then calculate the perimeter.
In a 30°-60°-90° triangle, the ratio of the sides is given by:
Shorter leg : Longer leg : Hypotenuse = 1 : √3 : 2
Since the length of the shorter leg is given as 2.5 feet, we can calculate the lengths of the other sides using the ratio.
1. To find the longer leg:
Let x be the length of the longer leg.
According to the ratio, we have:
2.5 : x : hypotenuse = 1 : √3 : 2
Cross multiplying, we get:
2.5 * 2 = x * √3
Simplifying, we have:
5 = x * √3
To isolate x, divide both sides by √3:
x = 5 / √3
2. To find the hypotenuse:
Using the same ratio, we have:
2.5 : longer leg : hypotenuse = 1 : √3 : 2
Cross multiplying, we get:
2.5 * 2 = √3 * hypotenuse
Simplifying, we have:
5 = √3 * hypotenuse
To isolate hypotenuse, divide both sides by √3:
hypotenuse = 5 / √3
Now that we have found the lengths of the longer leg and the hypotenuse, we can calculate the perimeter.
Perimeter = shorter leg + longer leg + hypotenuse
Perimeter = 2.5 + (5 / √3) + (5 / √3)
To simplify, rationalize the denominator:
Perimeter = 2.5 + (5√3 / 3) + (5√3 / 3)
Combine like terms:
Perimeter = 2.5 + (10√3 / 3) + (10√3 / 3)
Convert the fractions into a common denominator:
Perimeter = (7.5 / 3) + (10√3 / 3) + (10√3 / 3)
Add the fractions:
Perimeter = (7.5 + 10√3 + 10√3) / 3
Simplify and calculate:
Perimeter ≈ 11.83 feet
Therefore, the perimeter of the 30°-60°-90° triangle with a shorter leg of 2.5 feet is approximately 11.83 feet.