how do i find the side length of a right triangle with 105 m and 208 m

Answer 1

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!

Answer 2

now find the area of the rectangle

Answer 3

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.

Answer 4

how do i find the side length of a right triangle with 120 m and 210 m

Answer 5

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.

Answer 6

Now find the area of the triangle

Answer 7

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.