Question
What is the area of this triangle? Round your answer to the hundredths place.
Recall that area of a triangle is where b is the base and h is the height. The angle is 68 and the hypotenuse is 14 and A triangle has side lengths of square root of 3, 3 and 2 square root of 3. this is a right triangle because
Recall that area of a triangle is where b is the base and h is the height. The angle is 68 and the hypotenuse is 14 and A triangle has side lengths of square root of 3, 3 and 2 square root of 3. this is a right triangle because
Answers
GPT-4o mini
To find the area of the triangle, we can utilize the formula \( \text{Area} = \frac{1}{2} \times b \times h \), where \( b \) is the base and \( h \) is the height. However, since we have a right triangle indicated, we can use the side lengths of the triangle to determine the area.
Given the side lengths are \( \sqrt{3} \), \( 3 \), and \( 2\sqrt{3} \), we can identify that indeed it is a right triangle. The side lengths must adhere to the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides:
1. Identify the hypotenuse: The longest side, which we will consider is \( 3 \).
2. Check the other sides:
- \( (\sqrt{3})^2 + (2\sqrt{3})^2 = 3 + 12 = 15 \)
- \( (3)^2 = 9 \)
Since \( 3 \) is not the hypotenuse, let's take a look:
3. Hence, examining \( 2\sqrt{3} \):
- \( (2\sqrt{3})^2 = 12 \)
- Thus, verify:
- \( (\sqrt{3})^2 + 3^2 = 3 + 9 = 12 \), which indicates \( 2\sqrt{3} \) is valid as hypotenuse, confirming the triangle is indeed a right triangle.
Now, since this triangle has legs:
- \( a = \sqrt{3} \) (one leg)
- \( b = 3 \) (another leg)
The area can then be computed:
\[
\text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times \sqrt{3} \times 3
\]
Calculating:
\[
\text{Area} = \frac{3\sqrt{3}}{2}
\]
Next, we evaluate \( \frac{3\sqrt{3}}{2} \):
First, calculate \( \sqrt{3} \) (approximately \( 1.732 \)):
\[
3 \sqrt{3} \approx 3 \times 1.732 \approx 5.196
\]
Thus:
\[
\frac{5.196}{2} \approx 2.598
\]
Finally, rounding \( 2.598 \) to the hundredths place gives us:
\[
\text{Area} \approx 2.60
\]
Thus, the area of the triangle is approximately \( \boxed{2.60} \).
Given the side lengths are \( \sqrt{3} \), \( 3 \), and \( 2\sqrt{3} \), we can identify that indeed it is a right triangle. The side lengths must adhere to the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides:
1. Identify the hypotenuse: The longest side, which we will consider is \( 3 \).
2. Check the other sides:
- \( (\sqrt{3})^2 + (2\sqrt{3})^2 = 3 + 12 = 15 \)
- \( (3)^2 = 9 \)
Since \( 3 \) is not the hypotenuse, let's take a look:
3. Hence, examining \( 2\sqrt{3} \):
- \( (2\sqrt{3})^2 = 12 \)
- Thus, verify:
- \( (\sqrt{3})^2 + 3^2 = 3 + 9 = 12 \), which indicates \( 2\sqrt{3} \) is valid as hypotenuse, confirming the triangle is indeed a right triangle.
Now, since this triangle has legs:
- \( a = \sqrt{3} \) (one leg)
- \( b = 3 \) (another leg)
The area can then be computed:
\[
\text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times \sqrt{3} \times 3
\]
Calculating:
\[
\text{Area} = \frac{3\sqrt{3}}{2}
\]
Next, we evaluate \( \frac{3\sqrt{3}}{2} \):
First, calculate \( \sqrt{3} \) (approximately \( 1.732 \)):
\[
3 \sqrt{3} \approx 3 \times 1.732 \approx 5.196
\]
Thus:
\[
\frac{5.196}{2} \approx 2.598
\]
Finally, rounding \( 2.598 \) to the hundredths place gives us:
\[
\text{Area} \approx 2.60
\]
Thus, the area of the triangle is approximately \( \boxed{2.60} \).