Find the length of the diagonal of the prism.

Question

the height is 12 the length is 3 and the width is 4

Answer 1

To find the length of the diagonal of a rectangular prism, you can use the formula:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where:

• \( d \) is the length of the diagonal,
• \( l \) is the length,
• \( w \) is the width,
• \( h \) is the height.

Given:

• \( l = 3 \)
• \( w = 4 \)
• \( h = 12 \)

Now plug in the values:

\[ d = \sqrt{3^2 + 4^2 + 12^2} \] \[ d = \sqrt{9 + 16 + 144} \] \[ d = \sqrt{169} \] \[ d = 13 \]

Thus, the length of the diagonal of the prism is \( 13 \) units.

Answer 2

The Great Pyramid of Giza in Egypt has a square base with edge length of 230.5 m and a slant height of 180.2 m. Find the height of the pyramid. Round the answer to the nearest tenth

Answer 3

To find the height of the Great Pyramid of Giza, we can use the Pythagorean theorem. The height, the half of the base edge length, and the slant height form a right triangle.

1. First, calculate half of the base edge length: \[ \text{Half base length} = \frac{\text{Edge length}}{2} = \frac{230.5 , \text{m}}{2} = 115.25 , \text{m} \]

2. We denote the height of the pyramid as \( h \), the slant height as \( l \) (which is 180.2 m), and the half base length as \( b \) (which is 115.25 m).

The relationship between these can be expressed as: \[ l^2 = h^2 + b^2 \]

3. Plugging in the known values: \[ (180.2)^2 = h^2 + (115.25)^2 \]

4. Calculate \( (180.2)^2 \) and \( (115.25)^2 \): \[ (180.2)^2 = 32472.04 \] \[ (115.25)^2 = 13276.5625 \]

5. Now substitute back into the equation: \[ 32472.04 = h^2 + 13276.5625 \]

6. Solve for \( h^2 \): \[ h^2 = 32472.04 - 13276.5625 = 19195.4775 \]

7. Finally, take the square root to find \( h \): \[ h = \sqrt{19195.4775} \approx 138.6 , \text{m} \]

Rounding to the nearest tenth, the height of the Great Pyramid of Giza is approximately 138.6 meters.

Answer 4

Keysha is building a coffee table. It’s going to be in the shape of a rectangular prism, without the faces. She wants to make the table decorative by adding metal diagonals. The length of the coffee table will be 44 in., the width 24 in., and the height 14 in. Find the length Keysha should make each diagonal. Round the answer to the nearest tenth

Answer 5

To find the length of the diagonal of a rectangular prism, we can use the three-dimensional version of the Pythagorean theorem:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

Where:

• \( d \) is the length of the diagonal,
• \( l \) is the length,
• \( w \) is the width,
• \( h \) is the height.

Given:

• \( l = 44 \) in.
• \( w = 24 \) in.
• \( h = 14 \) in.

Now plug in the values:

\[ d = \sqrt{44^2 + 24^2 + 14^2} \] \[ d = \sqrt{1936 + 576 + 196} \] \[ d = \sqrt{2708} \]

Now, calculate \( \sqrt{2708} \):

\[ d \approx 51.9 , \text{in.} \quad (\text{after calculating the square root}) \]

Thus, the length of each diagonal Keysha should make is approximately 51.9 inches.

Answer 6

Explain a proof of the Pythagorean Theorem using the image.
4 right triangles touching each other form and open space that looks like a sideways square

Answer 7

To explain a proof of the Pythagorean Theorem using the configuration of four right triangles arranged around a square area, we can use a classic geometric approach. Although I can't display images, I can describe the scenario that leads to the proof.

Configuration

1. Create a Square: Start by constructing a large square of side length \( (a + b) \), where \( a \) and \( b \) are the legs of a right triangle.

2. Four Right Triangles: Place four identical right triangles in the corners of the square. Each triangle has legs of lengths \( a \) and \( b \) and a hypotenuse of length \( c \).

3. Inner Square Formation: The right triangles will touch each other at their hypotenuses, forming a smaller square in the center of the large square. The side length of this smaller inner square can be determined by the legs of the triangles.

Steps of the Proof

1. Calculate the Area of the Large Square: \[ \text{Area of Large Square} = (a + b)^2 \]

2. Calculate the Area of One Triangle: \[ \text{Area of One Triangle} = \frac{1}{2}ab \]

Since there are four triangles, the total area of the triangles is: \[ \text{Area of Four Triangles} = 4 \times \left(\frac{1}{2}ab\right) = 2ab \]

3. Area of the Inner Square: The side of the inner square is \( c \) (the hypotenuse of the triangles), so: \[ \text{Area of Inner Square} = c^2 \]

4. Express the Area Relationships: The area of the large square can also be expressed as the area of the four triangles plus the area of the inner square: \[ (a + b)^2 = \text{Area of Four Triangles} + \text{Area of Inner Square} \] \[ (a + b)^2 = 2ab + c^2 \]

5. Expand and Simplify: \[ a^2 + 2ab + b^2 = 2ab + c^2 \]

6. Cancel \( 2ab \) from both sides: \[ a^2 + b^2 = c^2 \]

Conclusion

Thus, we have derived the Pythagorean Theorem \( a^2 + b^2 = c^2 \) using the arrangement of four right triangles and the spaces they create. This geometric visualization not only illustrates the theorem mathematically but also provides a clear visual understanding of how the sides relate to each other in a right triangle.