The following statements about 30 – 60 – 90 triangles are true EXCEPT:
a. The shorter leg is the hypotenuse or times the longer leg.
b. The longer leg is times the shorter leg.
c. The longer leg is 3 times the shorter leg.
d. The hypotenuse is twice the shorter leg.
The correct answer and explanation is:
The correct answer is c. The longer leg is 3 times the shorter leg.
In a 30-60-90 triangle, the ratio of the lengths of the sides follows a specific pattern derived from the properties of right triangles. Here’s how it works:
- Hypotenuse: The hypotenuse is always twice the length of the shorter leg. This is a fundamental property of 30-60-90 triangles.
- Shorter Leg: The shorter leg is opposite the 30-degree angle and is typically denoted as x.
- Longer Leg: The longer leg is opposite the 60-degree angle and is denoted as x√3. This leg is longer than the shorter leg but not three times as long, which is why statement c is false.
To summarize, the correct relationships are:
- The hypotenuse = 2 × shorter leg (x).
- The longer leg = shorter leg × √3 (x√3).
- The shorter leg is simply x.
For example, if the shorter leg (opposite the 30-degree angle) is 5 units, the hypotenuse will be 10 units (5 × 2), and the longer leg will be 5√3, or approximately 8.66 units (5 × √3).
In this case, the statement that the longer leg is 3 times the shorter leg (c) does not hold true because the ratio of the longer leg to the shorter leg is actually √3, not 3. This means the longest side is not three times as long as the shortest side in a 30-60-90 triangle.