In a 30°-60°-90° triangle, if the hypotenuse is 20 km, find the exact values of the lengths of the legs.
the side ratio is ... 1 : √3 : 2
10 : 10√3 : 20
To find the exact values of the lengths of the legs in a 30°-60°-90° triangle with a hypotenuse of 20 km, we can use the ratio of the side lengths in this special triangle.
In a 30°-60°-90° triangle, the ratios of the side lengths are as follows:
- The length of the hypotenuse is always 2 times the length of the shorter leg.
- The length of the longer leg is always √3 times the length of the shorter leg.
Let's represent the length of the shorter leg as "x". Therefore, the length of the hypotenuse is 2x, and the length of the longer leg is √3x.
Given that the hypotenuse is 20 km, we have:
2x = 20
Simplifying the equation, we find:
x = 10
Therefore, the length of the shorter leg is 10 km, the length of the longer leg is √3 * 10 = 10√3 km, and the length of the hypotenuse is 20 km.
To find the lengths of the legs in a 30°-60°-90° triangle, we can use the ratios of the sides in this special type of triangle.
In a 30°-60°-90° triangle, the ratios of the sides are as follows:
- The length of the shorter leg is half the length of the hypotenuse.
- The length of the longer leg is √3 times the length of the shorter leg.
Given that the hypotenuse is 20 km, we can use these ratios to find the lengths of the legs:
1. Finding the shorter leg:
The length of the shorter leg is half the length of the hypotenuse, so:
Shorter leg = (1/2) * hypotenuse
Shorter leg = (1/2) * 20 km
Shorter leg = 10 km
2. Finding the longer leg:
The length of the longer leg is √3 times the length of the shorter leg, so:
Longer leg = √3 * shorter leg
Longer leg = √3 * 10 km
Longer leg = 10√3 km
Therefore, in a 30°-60°-90° triangle with a hypotenuse of 20 km, the exact lengths of the legs are 10 km and 10√3 km, respectively.