Question
First, we will use side lengths
in the Pythagorean Theorem to find the diagonal length of the base rectangle. Second, we will use 12 and the length we found in step one to solve for the diagonal of the prism. The length of the diagonal of the prism is
13
.
Hint: Don't forget to use your Desmos calculator and STAAR Reference Sheet to help!
in the Pythagorean Theorem to find the diagonal length of the base rectangle. Second, we will use 12 and the length we found in step one to solve for the diagonal of the prism. The length of the diagonal of the prism is
13
.
Hint: Don't forget to use your Desmos calculator and STAAR Reference Sheet to help!
Answers
GPT-4o mini
To solve for the diagonal of the prism using the information provided and the Pythagorean Theorem, we can break this down into two steps as suggested.
### Step 1: Find the diagonal length of the base rectangle.
Let's denote the side lengths of the rectangle as \( a \) and \( b \). The diagonal \( d \) of the rectangle can be calculated using the Pythagorean Theorem:
\[
d = \sqrt{a^2 + b^2}
\]
### Step 2: Use the diagonal of the rectangle and another dimension of the prism to find the diagonal of the prism.
The diagonal of a rectangular prism can also be calculated using the Pythagorean Theorem. If the height of the prism is \( h \) (which in this case is given as 12), and we have found the diagonal of the base rectangle \( d \), then the diagonal \( D \) of the prism can be found with:
\[
D = \sqrt{d^2 + h^2}
\]
### Given:
- You found that the diagonal of the prism \( D = 13 \).
- Height \( h = 12 \).
We can set up the equation:
\[
13 = \sqrt{d^2 + 12^2}
\]
### Solving for the diagonal of the base rectangle \( d \):
1. Simplify \( 12^2 \):
\[
12^2 = 144
\]
2. Substitute into the equation and square both sides:
\[
13^2 = d^2 + 144
\]
\[
169 = d^2 + 144
\]
3. Subtract 144 from both sides:
\[
d^2 = 169 - 144
\]
\[
d^2 = 25
\]
4. Take the square root:
\[
d = 5
\]
### Conclusion:
The diagonal length of the base rectangle is 5. If you know the dimensions of the base rectangle, you can verify that these dimensions also yield this diagonal using the formula \( d = \sqrt{a^2 + b^2} \). Please let me know if you need further assistance or specific values for \( a \) and \( b \)!
### Step 1: Find the diagonal length of the base rectangle.
Let's denote the side lengths of the rectangle as \( a \) and \( b \). The diagonal \( d \) of the rectangle can be calculated using the Pythagorean Theorem:
\[
d = \sqrt{a^2 + b^2}
\]
### Step 2: Use the diagonal of the rectangle and another dimension of the prism to find the diagonal of the prism.
The diagonal of a rectangular prism can also be calculated using the Pythagorean Theorem. If the height of the prism is \( h \) (which in this case is given as 12), and we have found the diagonal of the base rectangle \( d \), then the diagonal \( D \) of the prism can be found with:
\[
D = \sqrt{d^2 + h^2}
\]
### Given:
- You found that the diagonal of the prism \( D = 13 \).
- Height \( h = 12 \).
We can set up the equation:
\[
13 = \sqrt{d^2 + 12^2}
\]
### Solving for the diagonal of the base rectangle \( d \):
1. Simplify \( 12^2 \):
\[
12^2 = 144
\]
2. Substitute into the equation and square both sides:
\[
13^2 = d^2 + 144
\]
\[
169 = d^2 + 144
\]
3. Subtract 144 from both sides:
\[
d^2 = 169 - 144
\]
\[
d^2 = 25
\]
4. Take the square root:
\[
d = 5
\]
### Conclusion:
The diagonal length of the base rectangle is 5. If you know the dimensions of the base rectangle, you can verify that these dimensions also yield this diagonal using the formula \( d = \sqrt{a^2 + b^2} \). Please let me know if you need further assistance or specific values for \( a \) and \( b \)!