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Use the image to answer the question.

A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 9.

Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth.

(1 point)
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11 answers

To find the length between two points using the Pythagorean Theorem, we need the coordinates of the points. Assuming the points are \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \( d \) between these two points can be calculated using the formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

  1. First, identify the coordinates of the two points on the graph.
  2. Substitute the coordinates into the formula.
  3. Simplify the expression under the square root.
  4. Take the square root.
  5. Round the result to the nearest hundredth.

If you can provide the specific coordinates of the two points from the graph, I can help you calculate the distance.

the top one is 7,8 and boyyom is 3,2

To calculate the distance between the points \((7, 8)\) and \((3, 2)\) using the Pythagorean Theorem, we can apply the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Where:

  • \((x_1, y_1) = (7, 8)\)
  • \((x_2, y_2) = (3, 2)\)

Plugging in these values:

\[ d = \sqrt{(3 - 7)^2 + (2 - 8)^2} \]

Calculating the differences:

\[ d = \sqrt{(-4)^2 + (-6)^2} \]

Calculating the squares:

\[ d = \sqrt{16 + 36} \]

Adding these values:

\[ d = \sqrt{52} \]

Now, calculate the square root:

\[ d \approx 7.211 \]

Finally, rounding to the nearest hundredth:

\[ d \approx 7.21 \]

Thus, the length between the two points is approximately 7.21 units.

Use the image to answer the question.

An illustration shows a square pyramid with labeled parts. The front and left faces of the pyramid and base are visible. The back and right faces are not visible. The back and right base edges are not visible but their position is indicated by a dashed line. The left base edge shows a length of 16 feet. A dotted line extends from the interior of the vertex straight down to the center of the base and shows a height of 15. A right angle is indicated where the dotted height line meets the base.

Find the slant height of the square pyramid.

(1 point)
units

To find the slant height of the square pyramid, we need to use the relationship between the height of the pyramid, the half-length of the base, and the slant height itself.

  1. Identify the lengths:

    • The full length of the base of the pyramid is given as 16 feet.
    • The height from the vertex of the pyramid down to the center of the base is 15 feet.

  2. Calculate the half-length of the base: \[ \text{Half-length of the base} = \frac{16}{2} = 8 \text{ feet} \]

  3. Use the Pythagorean theorem to find the slant height (\( l \)). The slant height, the height of the pyramid, and the half-length of the base form a right triangle:

\[ l^2 = \text{height}^2 + \left(\text{half-length of the base}\right)^2 \]

Substituting the known values:

\[ l^2 = 15^2 + 8^2 \] \[ l^2 = 225 + 64 \] \[ l^2 = 289 \]

  1. Take the square root to find the slant height:

\[ l = \sqrt{289} = 17 \text{ feet} \]

Thus, the slant height of the square pyramid is 17 feet.

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.(1 point)
in.

To find the length of the diagonal of the rectangular prism (the coffee table), we can use the 3D distance formula. The diagonal \(d\) of a rectangular prism can be calculated using the formula:

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

where:

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

Given:

  • Length (\(l\)) = 44 inches
  • Width (\(w\)) = 24 inches
  • Height (\(h\)) = 14 inches

We can substitute these values into the formula:

\[ d = \sqrt{(44)^2 + (24)^2 + (14)^2} \]

Calculating each term:

\[ (44)^2 = 1936 \] \[ (24)^2 = 576 \] \[ (14)^2 = 196 \]

Now sum these values:

\[ d = \sqrt{1936 + 576 + 196} \] \[ d = \sqrt{2708} \]

Next, we calculate the square root:

\[ d \approx 52.0 \]

Thus, rounding to the nearest tenth, the length of each diagonal Keysha should make is approximately 52.0 inches.

Use the image to answer the question.

A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.

Explain a proof of the Pythagorean Theorem using the image.

(4 points)

To explain a proof of the Pythagorean Theorem using the described image, we can break down the reasoning into clear, logical steps based on the geometric relationships presented in the arrangement of the square and the right-angled triangles. Here's a structured explanation:

Step 1: Understanding the Setup

  1. Shape and Dimensions:

    • There is an outer square formed by the four right-angled triangles. Each triangle has height \(a\), base \(b\), and hypotenuse \(c\).
    • The inner square formed by the arrangement of these triangles is where the vertices of each triangle meet.

  2. Division of Sides:

    • Each side of the outer square is divided into two parts by the hypotenuses of the triangles: a shorter segment \(a\) and a longer segment \(b\), running along the side of the square.

Step 2: Establishing Relationships

  1. Relationship of Triangles to Square Sides:

    • When you place the right-angled triangles in the corners of the square, you introduce the segments along the sides of the square:
      • The total side length of the outer square is \( (a + b) + (a + b) = 2(a + b) \).

  2. Forming the Inner Square:

    • The vertices of the inner square are created by the intersection points of the hypotenuse \(c\) from each triangle. The sides of the inner square correspond to the length of \(c\).

Step 3: Applying the Pythagorean Theorem

  1. Triangles Within the Configuration:

    • Each right triangle adheres to the Pythagorean theorem, which states that for any right-angled triangle with legs \(a\) and \(b\), and hypotenuse \(c\): \[ c^2 = a^2 + b^2 \]

  2. Relating Hypotenuses to Squares:

    • The inner square's side \(c\) composes the larger square’s setup:
      • When you consider the area of the smaller inner square, it can also be viewed based on the right triangles that make up its corners and relates back to the properties of the larger square.

Step 4: Deriving the Proof

  1. Area Comparisons:

    • The area of the outer square (side length \(2a + 2b\)) is: \[ (2(a+b))^2 = 4(a+b)^2 = 4(a^2 + 2ab + b^2) \]

  2. Area of Inner Square:

    • The area of the inner square is simply \(c^2\).

  3. Infer Equivalence:

    • The combined areas of the four right triangles can be represented as: \[ \text{Area of the outer square} - \text{Area of the inner square} = \text{total area of triangles} \]
    • Thus, we see by the above relation that: \[ c^2 = a^2 + b^2 \]

Conclusion

In summary, the arrangement of parts in the figure shows that the areas and lengths support the equality \(c^2 = a^2 + b^2\), thus providing a conceptual proof of the Pythagorean theorem grounded in the geometric properties of the squares and right triangles. By demonstrating that the inner geometry relates back to the traditional triangle definitions, we see how the theorem holds true through this visual and spatial reasoning.

short it without extra details and make look like a 8th grader worit it

Proof of the Pythagorean Theorem Using the Square and Triangles

  1. Setup: In the image, there's a big square made from four right triangles at the corners. Each triangle has a height \(a\), a base \(b\), and a hypotenuse \(c\). The sides of the big square are made up of these triangles.

  2. Triangle Relationships: Each triangle follows the Pythagorean Theorem, which says that for a right triangle: \[ c^2 = a^2 + b^2 \]

  3. Inner Square: The hypotenuse \(c\) also acts as the side of a smaller square formed in the center. This smaller square shows us the connection between the triangles and the overall square shape.

  4. Area Comparison: The area of the large square can be found by looking at the areas of the triangles and the smaller square. The areas show us that: \[ c^2 = a^2 + b^2 \]

Conclusion

So, by seeing how the triangles fit inside the square and using their dimensions, we can prove the Pythagorean Theorem: \(c^2 = a^2 + b^2\).