Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine which statements about a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle are true or false, let's analyze each one considering the known properties of such a triangle.
### Properties of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] Triangle
In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
1. The hypotenuse is the longest side.
2. The length of the hypotenuse is twice the length of the shorter leg.
3. The length of the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Given this information, let's evaluate each statement:
#### Statement A:
The hypotenuse is twice as long as the shorter leg.
This is true. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the hypotenuse is indeed twice the length of the shorter leg.
#### Statement B:
The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
This is true. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
#### Statement C:
The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
This is false. The hypotenuse is twice as long as the shorter leg, not [tex]\(\sqrt{3}\)[/tex] times as long.
#### Statement D:
The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
This is false. The hypotenuse is not [tex]\(\sqrt{3}\)[/tex] times the length of the longer leg.
#### Statement E:
The hypotenuse is twice as long as the longer leg.
This is false. The hypotenuse is not twice the length of the longer leg; it is twice the length of the shorter leg.
#### Statement F:
The longer leg is twice as long as the shorter leg.
This is false. The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg, not twice.
### Conclusion
The true statements about a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle are:
- A. The hypotenuse is twice as long as the shorter leg.
- B. The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
The false statements are:
- C. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- D. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
- E. The hypotenuse is twice as long as the longer leg.
- F. The longer leg is twice as long as the shorter leg.
Hence, the final result is:
[tex]\[ (\text{{True Statements: }} \text{{['A', 'B']}}, \text{{False Statements: }} \text{{['C', 'D', 'E', 'F']}}) \][/tex]
### Properties of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] Triangle
In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
1. The hypotenuse is the longest side.
2. The length of the hypotenuse is twice the length of the shorter leg.
3. The length of the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Given this information, let's evaluate each statement:
#### Statement A:
The hypotenuse is twice as long as the shorter leg.
This is true. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the hypotenuse is indeed twice the length of the shorter leg.
#### Statement B:
The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
This is true. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
#### Statement C:
The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
This is false. The hypotenuse is twice as long as the shorter leg, not [tex]\(\sqrt{3}\)[/tex] times as long.
#### Statement D:
The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
This is false. The hypotenuse is not [tex]\(\sqrt{3}\)[/tex] times the length of the longer leg.
#### Statement E:
The hypotenuse is twice as long as the longer leg.
This is false. The hypotenuse is not twice the length of the longer leg; it is twice the length of the shorter leg.
#### Statement F:
The longer leg is twice as long as the shorter leg.
This is false. The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg, not twice.
### Conclusion
The true statements about a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle are:
- A. The hypotenuse is twice as long as the shorter leg.
- B. The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
The false statements are:
- C. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.
- D. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.
- E. The hypotenuse is twice as long as the longer leg.
- F. The longer leg is twice as long as the shorter leg.
Hence, the final result is:
[tex]\[ (\text{{True Statements: }} \text{{['A', 'B']}}, \text{{False Statements: }} \text{{['C', 'D', 'E', 'F']}}) \][/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
