IDNLearn.com makes it easy to get reliable answers from experts and enthusiasts alike. Ask anything and receive comprehensive, well-informed responses from our dedicated team of experts.
Sagot :
To find the length of the hypotenuse in the right triangle given the shortest side and one of the angles, follow these steps:
1. Identify the type of right triangle:
The triangle described has a shortest side and a [tex]$60^{\circ}$[/tex] angle. This is a 30-60-90 right triangle, where the angles are [tex]$30^{\circ}$[/tex], [tex]$60^{\circ}$[/tex], and [tex]$90^{\circ}$[/tex].
2. Understand the side ratios of a 30-60-90 right triangle:
In this type of triangle, the sides are in a specific ratio:
- The side opposite the [tex]$30^{\circ}$[/tex] angle (shortest side) is [tex]\( x \)[/tex].
- The side opposite the [tex]$60^{\circ}$[/tex] angle (longer leg) is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2x \)[/tex].
3. Given:
The shortest side is [tex]\( 3 \sqrt{3} \)[/tex] inches. This side is opposite the [tex]$30^{\circ}$[/tex] angle.
4. Set up the ratio and solve for the hypotenuse:
- The shortest side (opposite the [tex]$30^{\circ}$[/tex] angle) is [tex]\( x \)[/tex], so [tex]\( x = 3 \sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2x \)[/tex].
5. Calculate the hypotenuse:
[tex]\[ \text{Hypotenuse} = 2x = 2 \cdot (3\sqrt{3}) = 6\sqrt{3} \][/tex]
So, we have calculated that the length of the hypotenuse is equal to [tex]\( 6 \sqrt{3} \)[/tex] inches.
6. Check the options:
A. [tex]\( 6 \sqrt{2} \)[/tex]
B. 6
C. 3
D. [tex]\( 6 \sqrt{3} \)[/tex]
The correct answer is:
[tex]\[ \boxed{6} \][/tex]
1. Identify the type of right triangle:
The triangle described has a shortest side and a [tex]$60^{\circ}$[/tex] angle. This is a 30-60-90 right triangle, where the angles are [tex]$30^{\circ}$[/tex], [tex]$60^{\circ}$[/tex], and [tex]$90^{\circ}$[/tex].
2. Understand the side ratios of a 30-60-90 right triangle:
In this type of triangle, the sides are in a specific ratio:
- The side opposite the [tex]$30^{\circ}$[/tex] angle (shortest side) is [tex]\( x \)[/tex].
- The side opposite the [tex]$60^{\circ}$[/tex] angle (longer leg) is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2x \)[/tex].
3. Given:
The shortest side is [tex]\( 3 \sqrt{3} \)[/tex] inches. This side is opposite the [tex]$30^{\circ}$[/tex] angle.
4. Set up the ratio and solve for the hypotenuse:
- The shortest side (opposite the [tex]$30^{\circ}$[/tex] angle) is [tex]\( x \)[/tex], so [tex]\( x = 3 \sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2x \)[/tex].
5. Calculate the hypotenuse:
[tex]\[ \text{Hypotenuse} = 2x = 2 \cdot (3\sqrt{3}) = 6\sqrt{3} \][/tex]
So, we have calculated that the length of the hypotenuse is equal to [tex]\( 6 \sqrt{3} \)[/tex] inches.
6. Check the options:
A. [tex]\( 6 \sqrt{2} \)[/tex]
B. 6
C. 3
D. [tex]\( 6 \sqrt{3} \)[/tex]
The correct answer is:
[tex]\[ \boxed{6} \][/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.