Asked by gedagedigedagedago
how do i find the side length of a right triangle with 105 m and 208 m
Answers
Answered by
gedagedigedagedago
now find the area of the rectangle
Answered by
gedagedigedagedago
how do i find the side length of a right triangle with 120 m and 210 m
Answered by
gedagedigedagedago
Now find the area of the triangle
Answered by
GPT-4o mini
To find the side length of a right triangle when you have two sides, you can use the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse (the longest side), and \(a\) and \(b\) are the lengths of the other two sides.
Given:
- One side \(a = 105\) m
- Another side \(b = 208\) m
If these are the two legs (the sides adjacent to the right angle), you can find the hypotenuse \(c\):
\[
c = \sqrt{a^2 + b^2}
\]
Substituting in the values:
\[
c = \sqrt{105^2 + 208^2}
\]
Calculating the squares:
\[
105^2 = 11025
\]
\[
208^2 = 43264
\]
Now add them together:
\[
c = \sqrt{11025 + 43264}
\]
\[
c = \sqrt{54289}
\]
Now, calculate the square root:
\[
c \approx 233 \text{ m}
\]
So, if 105 m and 208 m are the legs, the hypotenuse is approximately **233 m**.
If instead, you need to find the length of one of the legs and you know it is the hypotenuse (let's say 208 m), then you can rearrange the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
If \(c = 208\) m, and we assume \(a = 105\) m:
\[
105^2 + b^2 = 208^2
\]
\[
11025 + b^2 = 43264
\]
\[
b^2 = 43264 - 11025
\]
\[
b^2 = 32239
\]
Taking the square root:
\[
b \approx \sqrt{32239} \approx 179.3 \text{ m}
\]
So, if 208 m is the hypotenuse and 105 m is one of the legs, the other leg would be approximately **179.3 m**. You can clarify which triangle dimensions you have if needed!
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse (the longest side), and \(a\) and \(b\) are the lengths of the other two sides.
Given:
- One side \(a = 105\) m
- Another side \(b = 208\) m
If these are the two legs (the sides adjacent to the right angle), you can find the hypotenuse \(c\):
\[
c = \sqrt{a^2 + b^2}
\]
Substituting in the values:
\[
c = \sqrt{105^2 + 208^2}
\]
Calculating the squares:
\[
105^2 = 11025
\]
\[
208^2 = 43264
\]
Now add them together:
\[
c = \sqrt{11025 + 43264}
\]
\[
c = \sqrt{54289}
\]
Now, calculate the square root:
\[
c \approx 233 \text{ m}
\]
So, if 105 m and 208 m are the legs, the hypotenuse is approximately **233 m**.
If instead, you need to find the length of one of the legs and you know it is the hypotenuse (let's say 208 m), then you can rearrange the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
If \(c = 208\) m, and we assume \(a = 105\) m:
\[
105^2 + b^2 = 208^2
\]
\[
11025 + b^2 = 43264
\]
\[
b^2 = 43264 - 11025
\]
\[
b^2 = 32239
\]
Taking the square root:
\[
b \approx \sqrt{32239} \approx 179.3 \text{ m}
\]
So, if 208 m is the hypotenuse and 105 m is one of the legs, the other leg would be approximately **179.3 m**. You can clarify which triangle dimensions you have if needed!
Answered by
GPT-4o mini
To find the area of a rectangle, you use the formula:
\[
\text{Area} = \text{length} \times \text{width}
\]
Assuming the sides of the rectangle are the values you provided, 105 m and 208 m, we can calculate the area as follows:
\[
\text{Area} = 105 \, \text{m} \times 208 \, \text{m}
\]
Calculating that gives:
\[
\text{Area} = 21840 \, \text{m}^2
\]
Therefore, the area of the rectangle is **21,840 square meters**.
\[
\text{Area} = \text{length} \times \text{width}
\]
Assuming the sides of the rectangle are the values you provided, 105 m and 208 m, we can calculate the area as follows:
\[
\text{Area} = 105 \, \text{m} \times 208 \, \text{m}
\]
Calculating that gives:
\[
\text{Area} = 21840 \, \text{m}^2
\]
Therefore, the area of the rectangle is **21,840 square meters**.
Answered by
GPT-4o mini
To find the side length of a right triangle with the sides measuring 120 m and 210 m, you can again use the Pythagorean theorem, which is given by:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse (the longest side), and \(a\) and \(b\) are the lengths of the legs of the triangle.
### Case 1: Finding the hypotenuse
If 120 m and 210 m are the lengths of the two legs \(a\) and \(b\):
\[
c = \sqrt{a^2 + b^2} = \sqrt{120^2 + 210^2}
\]
Calculating the squares:
\[
120^2 = 14400
\]
\[
210^2 = 44100
\]
Now add them:
\[
c = \sqrt{14400 + 44100} = \sqrt{58500}
\]
Calculating the square root:
\[
c \approx 241.8 \text{ m}
\]
So, if 120 m and 210 m are the legs, the length of the hypotenuse \(c\) is approximately **241.8 m**.
### Case 2: Finding a leg if one side is the hypotenuse
If one of the sides is the hypotenuse (let's assume 210 m is the hypotenuse), and you want to find the other leg \(b\) given \(a = 120\) m:
Using the rearranged Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
Thus,
\[
210^2 = 120^2 + b^2
\]
Calculating the squares:
\[
44100 = 14400 + b^2
\]
Now solve for \(b^2\):
\[
b^2 = 44100 - 14400
\]
\[
b^2 = 29700
\]
Now take the square root:
\[
b \approx \sqrt{29700} \approx 172.3 \text{ m}
\]
So, if 210 m is the hypotenuse and 120 m is one of the legs, the other leg \(b\) is approximately **172.3 m**.
### Summary:
- If both 120 m and 210 m are legs, then the hypotenuse is approximately **241.8 m**.
- If 210 m is the hypotenuse, then the length of the other leg is approximately **172.3 m**.
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse (the longest side), and \(a\) and \(b\) are the lengths of the legs of the triangle.
### Case 1: Finding the hypotenuse
If 120 m and 210 m are the lengths of the two legs \(a\) and \(b\):
\[
c = \sqrt{a^2 + b^2} = \sqrt{120^2 + 210^2}
\]
Calculating the squares:
\[
120^2 = 14400
\]
\[
210^2 = 44100
\]
Now add them:
\[
c = \sqrt{14400 + 44100} = \sqrt{58500}
\]
Calculating the square root:
\[
c \approx 241.8 \text{ m}
\]
So, if 120 m and 210 m are the legs, the length of the hypotenuse \(c\) is approximately **241.8 m**.
### Case 2: Finding a leg if one side is the hypotenuse
If one of the sides is the hypotenuse (let's assume 210 m is the hypotenuse), and you want to find the other leg \(b\) given \(a = 120\) m:
Using the rearranged Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
Thus,
\[
210^2 = 120^2 + b^2
\]
Calculating the squares:
\[
44100 = 14400 + b^2
\]
Now solve for \(b^2\):
\[
b^2 = 44100 - 14400
\]
\[
b^2 = 29700
\]
Now take the square root:
\[
b \approx \sqrt{29700} \approx 172.3 \text{ m}
\]
So, if 210 m is the hypotenuse and 120 m is one of the legs, the other leg \(b\) is approximately **172.3 m**.
### Summary:
- If both 120 m and 210 m are legs, then the hypotenuse is approximately **241.8 m**.
- If 210 m is the hypotenuse, then the length of the other leg is approximately **172.3 m**.
Answered by
GPT-4o mini
To find the area of a right triangle, you can use the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In a right triangle, the two legs serve as the base and the height. Based on your earlier mention of a triangle with sides of 120 m and 210 m, you can use these two lengths as the base and height.
Assuming that:
- Base \( = 120 \, \text{m} \)
- Height \( = 210 \, \text{m} \)
Now, plug these values into the area formula:
\[
\text{Area} = \frac{1}{2} \times 120 \, \text{m} \times 210 \, \text{m}
\]
Calculating it step-by-step:
1. Multiply the base and height:
\[
120 \times 210 = 25200 \, \text{m}^2
\]
2. Now, take half of that product:
\[
\text{Area} = \frac{1}{2} \times 25200 = 12600 \, \text{m}^2
\]
Thus, the area of the right triangle is **12,600 square meters**.
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In a right triangle, the two legs serve as the base and the height. Based on your earlier mention of a triangle with sides of 120 m and 210 m, you can use these two lengths as the base and height.
Assuming that:
- Base \( = 120 \, \text{m} \)
- Height \( = 210 \, \text{m} \)
Now, plug these values into the area formula:
\[
\text{Area} = \frac{1}{2} \times 120 \, \text{m} \times 210 \, \text{m}
\]
Calculating it step-by-step:
1. Multiply the base and height:
\[
120 \times 210 = 25200 \, \text{m}^2
\]
2. Now, take half of that product:
\[
\text{Area} = \frac{1}{2} \times 25200 = 12600 \, \text{m}^2
\]
Thus, the area of the right triangle is **12,600 square meters**.