Answer :
In a 30-60-90 triangle, there are certain proportional relationships between the lengths of the sides. The basic rules of a 30-60-90 triangle state that:
- The hypotenuse is twice the length of the shorter leg.
- The longer leg is [tex]\( \sqrt{3} \)[/tex] times the length of the shorter leg.
Let's evaluate each statement:
A. The longer leg is [tex]\( \sqrt{3} \)[/tex] times as long as the shorter leg.
- This is a true statement. The longer leg in a 30-60-90 triangle is indeed [tex]\( \sqrt{3} \)[/tex] times the shorter leg.
B. The hypotenuse is twice as long as the longer leg.
- This is a false statement. The hypotenuse is twice the length of the shorter leg, not the longer leg.
C. The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as the shorter leg.
- This is a false statement. The hypotenuse is twice the length of the shorter leg, not [tex]\( \sqrt{3} \)[/tex] times.
D. The longer leg is twice as long as the shorter leg.
- This is a false statement. The longer leg is [tex]\( \sqrt{3} \)[/tex] times the shorter leg, not twice.
E. The hypotenuse is twice as long as the shorter leg.
- This is a true statement. The hypotenuse in a 30-60-90 triangle is indeed twice the length of the shorter leg.
F. The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as the longer leg.
- This is a false statement. The correct relationship is that the longer leg is [tex]\( \sqrt{3} \)[/tex] times the shorter leg, not the hypotenuse.
So, the true statements about a 30-60-90 triangle are:
A. The longer leg is [tex]\( \sqrt{3} \)[/tex] times as long as the shorter leg.
E. The hypotenuse is twice as long as the shorter leg.
- The hypotenuse is twice the length of the shorter leg.
- The longer leg is [tex]\( \sqrt{3} \)[/tex] times the length of the shorter leg.
Let's evaluate each statement:
A. The longer leg is [tex]\( \sqrt{3} \)[/tex] times as long as the shorter leg.
- This is a true statement. The longer leg in a 30-60-90 triangle is indeed [tex]\( \sqrt{3} \)[/tex] times the shorter leg.
B. The hypotenuse is twice as long as the longer leg.
- This is a false statement. The hypotenuse is twice the length of the shorter leg, not the longer leg.
C. The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as the shorter leg.
- This is a false statement. The hypotenuse is twice the length of the shorter leg, not [tex]\( \sqrt{3} \)[/tex] times.
D. The longer leg is twice as long as the shorter leg.
- This is a false statement. The longer leg is [tex]\( \sqrt{3} \)[/tex] times the shorter leg, not twice.
E. The hypotenuse is twice as long as the shorter leg.
- This is a true statement. The hypotenuse in a 30-60-90 triangle is indeed twice the length of the shorter leg.
F. The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as the longer leg.
- This is a false statement. The correct relationship is that the longer leg is [tex]\( \sqrt{3} \)[/tex] times the shorter leg, not the hypotenuse.
So, the true statements about a 30-60-90 triangle are:
A. The longer leg is [tex]\( \sqrt{3} \)[/tex] times as long as the shorter leg.
E. The hypotenuse is twice as long as the shorter leg.
