Question
What is the lenght of the hypotenuse of a 30-60-90 triangle if the side opposite the 60-degree angle is 6 square root of 3?
Answers
GPT-4o mini
In a 30-60-90 triangle, the ratios of the lengths of the sides are as follows:
- The side opposite the 30-degree angle (let's call it \( a \)) is \( x \).
- The side opposite the 60-degree angle (let's call it \( b \)) is \( x\sqrt{3} \).
- The hypotenuse (let's call it \( c \)) is \( 2x \).
From your question, we know that the side opposite the 60-degree angle \( b \) is \( 6\sqrt{3} \). According to the properties of a 30-60-90 triangle, we can set up the equation:
\[
b = x\sqrt{3}
\]
Setting \( b = 6\sqrt{3} \), we have:
\[
x\sqrt{3} = 6\sqrt{3}
\]
To solve for \( x \), divide both sides by \( \sqrt{3} \):
\[
x = 6
\]
Now that \( x \) is known, we can find the length of the hypotenuse \( c \):
\[
c = 2x = 2(6) = 12
\]
Thus, the length of the hypotenuse is \( \boxed{12} \).
- The side opposite the 30-degree angle (let's call it \( a \)) is \( x \).
- The side opposite the 60-degree angle (let's call it \( b \)) is \( x\sqrt{3} \).
- The hypotenuse (let's call it \( c \)) is \( 2x \).
From your question, we know that the side opposite the 60-degree angle \( b \) is \( 6\sqrt{3} \). According to the properties of a 30-60-90 triangle, we can set up the equation:
\[
b = x\sqrt{3}
\]
Setting \( b = 6\sqrt{3} \), we have:
\[
x\sqrt{3} = 6\sqrt{3}
\]
To solve for \( x \), divide both sides by \( \sqrt{3} \):
\[
x = 6
\]
Now that \( x \) is known, we can find the length of the hypotenuse \( c \):
\[
c = 2x = 2(6) = 12
\]
Thus, the length of the hypotenuse is \( \boxed{12} \).
u sure
GPT-4o mini
Yes, let's confirm the steps to ensure the solution is correct.
In a 30-60-90 triangle, the relationships between the sides are as follows:
- The side opposite the 30-degree angle (let's call it \( a \)) is \( x \).
- The side opposite the 60-degree angle (let's call it \( b \)) is \( x\sqrt{3} \).
- The hypotenuse (let's call it \( c \)) is \( 2x \).
You stated that the side opposite the 60-degree angle is \( 6\sqrt{3} \). Therefore, we can set up the equation:
\[
b = x\sqrt{3}
\]
Substituting \( b = 6\sqrt{3} \), we have:
\[
x\sqrt{3} = 6\sqrt{3}
\]
To isolate \( x \), we divide both sides by \( \sqrt{3} \):
\[
x = 6
\]
Next, we find the length of the hypotenuse \( c \) using the relation \( c = 2x \):
\[
c = 2(6) = 12
\]
So, the length of the hypotenuse of the triangle is indeed \( \boxed{12} \). If you have any further questions or need additional clarification, feel free to ask!
In a 30-60-90 triangle, the relationships between the sides are as follows:
- The side opposite the 30-degree angle (let's call it \( a \)) is \( x \).
- The side opposite the 60-degree angle (let's call it \( b \)) is \( x\sqrt{3} \).
- The hypotenuse (let's call it \( c \)) is \( 2x \).
You stated that the side opposite the 60-degree angle is \( 6\sqrt{3} \). Therefore, we can set up the equation:
\[
b = x\sqrt{3}
\]
Substituting \( b = 6\sqrt{3} \), we have:
\[
x\sqrt{3} = 6\sqrt{3}
\]
To isolate \( x \), we divide both sides by \( \sqrt{3} \):
\[
x = 6
\]
Next, we find the length of the hypotenuse \( c \) using the relation \( c = 2x \):
\[
c = 2(6) = 12
\]
So, the length of the hypotenuse of the triangle is indeed \( \boxed{12} \). If you have any further questions or need additional clarification, feel free to ask!