Directions: Use this worksheet to record answers to the four activities that make

Question

up the Pythagorean Theorem Portfolio. When you are finished, save this worksheet
with your answers and submit it for a portfolio grade.
Finding the Area of a Field
Use the image to answer to complete the activity.
The image features a coordinate plane showcasing three fields: Field A, Field B, and Field C. Each field is represented as a blue square or diamond, with coordinates marked at their vertices. Here's a breakdown:

Field A: Vertices at (-4, 4), (0, 4), (0, 0), and (-4, 0)
Field B: Vertices at (0, 0), (3, 0), (3, -3), and (0, -3)
Field C: Vertices at (0, 0), (4, 7), (7, 3), and (3, 0)

1. Count the number of squares inside Field A. This is the area of Square A in
acres. Write it below:
Area = __________ acres
2. Count the number of squares inside Field B. This is the area of Square B in
acres. Write it below:
Area = __________ acres
3. Use the Pythagorean Theorem to find the area of Field C in acres:
Area = __________ acre

Answer 1

To find the areas of the fields based on the provided coordinates, let's calculate each one step by step.

1. Area of Field A

Vertices of Field A: (-4, 4), (0, 4), (0, 0), (-4, 0)

Field A forms a square. The length of one side can be determined by the distance between two vertices.

• The distance from (-4, 4) to (0, 4) is \(0 - (-4) = 4\).

Thus, the area \(A\) of Field A is: \[ A = \text{side}^2 = 4^2 = 16 \text{ square units (or acres)} \]

Answer:
Area = 16 acres

2. Area of Field B

Vertices of Field B: (0, 0), (3, 0), (3, -3), (0, -3)

Field B also forms a square. The length of one side can be determined in a similar manner.

• The distance from (0, 0) to (3, 0) is \(3 - 0 = 3\).

Thus, the area \(B\) of Field B is: \[ B = \text{side}^2 = 3^2 = 9 \text{ square units (or acres)} \]

Answer:
Area = 9 acres

3. Area of Field C

Vertices of Field C: (0, 0), (4, 7), (7, 3), (3, 0)

Field C does not form a square but a quadrilateral. To find the area, we can divide it into two triangles or apply the shoelace formula.

Let's use the shoelace formula: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \]

Substituting the coordinates into the formula where:

• \((x_1, y_1) = (0, 0)\)
• \((x_2, y_2) = (4, 7)\)
• \((x_3, y_3) = (7, 3)\)
• \((x_4, y_4) = (3, 0)\)

Calculating: \[ \text{Area} = \frac{1}{2} \left| 0 \cdot 7 + 4 \cdot 3 + 7 \cdot 0 + 3 \cdot 0 - (0 \cdot 4 + 7 \cdot 7 + 3 \cdot 3 + 0 \cdot 0) \right| \] \[ = \frac{1}{2} \left| 0 + 12 + 0 + 0 - (0 + 49 + 9 + 0) \right| \] \[ = \frac{1}{2} \left| 12 - 58 \right| \] \[ = \frac{1}{2} \left| -46 \right| = \frac{46}{2} = 23 \text{ square units (or acres)} \]

Answer:
Area = 23 acres

Summary of Areas:

1. Area of Field A = 16 acres
2. Area of Field B = 9 acres
3. Area of Field C = 23 acres

Answer 2

Finding the Area of a Garden
Use the image to complete the activity.
The image depicts a Cartesian coordinate plane labeled from -12 to 12 on both the x-axis and y-axis. The grid is made up of equal squares, each representing one unit. The origin (0,0) is marked at the center where the x-axis and y-axis intersect.
1. A square garden has the end points listed below. Plot and label these points on
the grid:
a. A (-7, 6)
b. B (-2, -6)
c. C (10, -1)
d. D (5, 11)
2. Connect the points to create Square ABCD.
3. Use the Pythagorean Theorem to find the side length, s, of Square ABCD in
feet.
s = ________feet
4. Use the area formula, 𝐴𝐴 = 𝑠𝑠2, to find the area of Square ABCD.
A = ________feet^2

Answer 3

To solve the problem regarding the square garden, we'll follow the instructions step by step.

Step 1: Plot Points A, B, C, and D

Given Points:

• A (-7, 6)
• B (-2, -6)
• C (10, -1)
• D (5, 11)

Locate and plot these points on the Cartesian coordinate plane:

• Point A (-7, 6) will be in the second quadrant.
• Point B (-2, -6) will be in the fourth quadrant.
• Point C (10, -1) will be in the fourth quadrant.
• Point D (5, 11) will be in the second quadrant.

Step 2: Connect the Points to Create Square ABCD

Since we were given four points, it seems they do not form a perfect square based on their coordinates. To determine the arrangement of these points, we can calculate the distances between the points and identify which should be connected to form the quadrilateral.

Step 3: Find the Side Length of Square ABCD using the Pythagorean Theorem

We will calculate the length of the sides one by one.

Finding Length of AB:

• A (-7, 6) to B (-2, -6): \[ d_{AB} = \sqrt{(-2 - (-7))^2 + (-6 - 6)^2} = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ feet} \]

Finding Length of BC:

• B (-2, -6) to C (10, -1): \[ d_{BC} = \sqrt{(10 - (-2))^2 + (-1 - (-6))^2} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ feet} \]

Finding Length of CD:

• C (10, -1) to D (5, 11): \[ d_{CD} = \sqrt{(5 - 10)^2 + (11 - (-1))^2} = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ feet} \]

Finding Length of DA:

• D (5, 11) to A (-7, 6): \[ d_{DA} = \sqrt{(-7 - 5)^2 + (6 - 11)^2} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ feet} \]

Conclusion for Side Length

All sides of Quadrilateral ABCD are 13 feet.

Answer:
s = 13 feet

Step 4: Find the Area of Square ABCD

Using the area formula \( A = s^2 \): \[ A = (13)^2 = 169 \text{ feet}^2 \]

Answer:
A = 169 feet²

Summary of Findings

1. Points plotted and connected to form Quadrilateral ABCD.
2. Side length s = 13 feet
3. Area A = 169 feet²

Answer 4

Computer screens and television screens are usually measured by the diagonal of
the screen. For example, a screen of size 60 inches would have a diagonal of 60
inches, a width of 52 inches, and a height of 30 inches.
Yoselin would like to install a new screen on a shelf that is 17 inches tall and 50
inches wide. She would like the screen to be the largest size that will fit on the
shelf. She knows the diagonal and width of the 4 screens listed below.

Screen Diagonal Width Height
A 15” 13”
B 22” 19”
C 32” 28”
D 40” 35”

1. Use the Pythagorean Theorem to calculate the height of each screen, and
complete the table.










2. Which screen is the largest that will fit on Yoselin’s shelf?

Answer 5

To find the height of each screen, we can use the Pythagorean Theorem, which states:

\[ a^2 + b^2 = c^2 \]

Where:

• \( a \) is the width,
• \( b \) is the height,
• \( c \) is the diagonal.

We will rearrange the formula to solve for the height \( b \):

\[ b = \sqrt{c^2 - a^2} \]

Now we can calculate the height for each screen.

Given Data:

1. Screen A:

   • Diagonal = 15 inches
   • Width = 13 inches

2. Screen B:

   • Diagonal = 22 inches
   • Width = 19 inches

3. Screen C:

   • Diagonal = 32 inches
   • Width = 28 inches

4. Screen D:

   • Diagonal = 40 inches
   • Width = 35 inches

Calculating Heights:

Screen A:

Diagonal = 15 inches
Width = 13 inches
\[ b = \sqrt{15^2 - 13^2} \]
\[ b = \sqrt{225 - 169} \]
\[ b = \sqrt{56} \approx 7.48 \text{ inches} \]

Screen B:

Diagonal = 22 inches
Width = 19 inches
\[ b = \sqrt{22^2 - 19^2} \]
\[ b = \sqrt{484 - 361} \]
\[ b = \sqrt{123} \approx 11.09 \text{ inches} \]

Screen C:

Diagonal = 32 inches
Width = 28 inches
\[ b = \sqrt{32^2 - 28^2} \]
\[ b = \sqrt{1024 - 784} \]
\[ b = \sqrt{240} \approx 15.49 \text{ inches} \]

Screen D:

Diagonal = 40 inches
Width = 35 inches
\[ b = \sqrt{40^2 - 35^2} \]
\[ b = \sqrt{1600 - 1225} \]
\[ b = \sqrt{375} \approx 19.36 \text{ inches} \]

Completing the Table:

| Screen | Diagonal | Width | Height (inches) | |--------|----------|-------|------------------| | A | 15" | 13" | 7.48 | | B | 22" | 19" | 11.09 | | C | 32" | 28" | 15.49 | | D | 40" | 35" | 19.36 |

Step 2: Determine the Largest Screen that Will Fit on Yoselin's Shelf

Yoselin's shelf dimensions:

• Height = 17 inches
• Width = 50 inches

To determine which screen will fit:

• The width and height of the screen must be less than or equal to 50 inches and 17 inches, respectively.

Fit Criteria:

1. Screen A: Height = 7.48" (Fits)
2. Screen B: Height = 11.09" (Fits)
3. Screen C: Height = 15.49" (Fits)
4. Screen D: Height = 19.36" (Does Not Fit)

Conclusion:

The largest screen that will fit on Yoselin’s shelf is Screen C, which has a height of approximately 15.49 inches and a diagonal of 32 inches.

Answer 6

why did they say Screen A: Height = 7.48 inches (Fits)
Screen B: Height = 11.09 inches (Fits)
Screen C: Height = 15.49 inches (Fits)
Screen D: Height = 19.36 inches (Does not fit)
Among the screens that fit, the largest screen in terms of diagonal is Screen C at 32 inches.

Thus, the largest screen that will fit on Yoselin’s shelf is Screen C (32 inches).

and someone else said Screen A: 9 inches

Screen B: 11 inches

Screen C: 17.89 inches

Screen D: 23.5 inches

The biggest screen that will fit on Yoselin’s shelf is Screen B.

Answer 7

Let's clarify the height calculations and the fitting criteria for the screens once again to determine the correct dimensions and identify the largest screen that fits on Yoselin’s shelf.

We will use the Pythagorean Theorem to calculate each screen's height based on the diagonal and width provided. Here's the method we used:

Pythagorean Theorem

For a screen, given:

• \( c \) is the diagonal,
• \( a \) is the width,
• \( b \) (which we want to find) is the height,

We have the formula: \[ b = \sqrt{c^2 - a^2} \]

Given Data for the Screens

1. Screen A:

   • Diagonal = 15 inches
   • Width = 13 inches

2. Screen B:

   • Diagonal = 22 inches
   • Width = 19 inches

3. Screen C:

   • Diagonal = 32 inches
   • Width = 28 inches

4. Screen D:

   • Diagonal = 40 inches
   • Width = 35 inches

Re-calculating Heights

Screen A:

\[ b_A = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.48 \text{ inches} \]

Screen B:

\[ b_B = \sqrt{22^2 - 19^2} = \sqrt{484 - 361} = \sqrt{123} \approx 11.09 \text{ inches} \]

Screen C:

\[ b_C = \sqrt{32^2 - 28^2} = \sqrt{1024 - 784} = \sqrt{240} \approx 15.49 \text{ inches} \]

Screen D:

\[ b_D = \sqrt{40^2 - 35^2} = \sqrt{1600 - 1225} = \sqrt{375} \approx 19.36 \text{ inches} \]

Summary of Heights:

| Screen | Diagonal | Width | Height (inches) | |--------|----------|-------|------------------| | A | 15" | 13" | 7.48 | | B | 22" | 19" | 11.09 | | C | 32" | 28" | 15.49 | | D | 40" | 35" | 19.36 |

Fit Criteria Based on Yoselin’s Shelf:

Yoselin's shelf dimensions:

• Height = 17 inches
• Width = 50 inches

The screens that fit:

1. Screen A: Height = 7.48 inches (Fits)
2. Screen B: Height = 11.09 inches (Fits)
3. Screen C: Height = 15.49 inches (Fits)
4. Screen D: Height = 19.36 inches (Does not fit)

Result Analysis:

Among the screens that fit:

• Screen A (7.48 inches) fits.
• Screen B (11.09 inches) fits.
• Screen C (15.49 inches) fits.
• Screen D (19.36 inches) does not fit since it exceeds 17 inches in height.

Hence, the largest screen in terms of diagonal among those that fit is Screen C (32 inches).

Addressing the Other Heights Mentioned:

The heights of 9 inches, 11 inches, 17.89 inches, and 23.5 inches mentioned by someone else seem to be incorrect and do not align with the calculations verified through the Pythagorean theorem. Based on the provided screen dimensions and my calculations, Screen B is not the largest available screen since its height is less than that of Screen C.

Conclusion:

From our calculation, the largest screen that will fit on Yoselin’s shelf is Screen C (32 inches diagonal, height approximately 15.49 inches).

Answer 8

The image shows a geometric diagram of a right triangle with points labeled D, A (Base of Stairs), B, and C (Top of Stairs). The distance from D to A is 30 inches, and the vertical distances for the steps are labeled as 7 inches and 11 inches. There are blank spaces for additional measurements along the hypotenuse and the horizontal and vertical sides of the triangle.

Answer 9

To calculate various measurements for the right triangle described in your geometry problem, we can utilize the Pythagorean Theorem and basic properties of right triangles.

Geometry of the Right Triangle

Let's define the points and their relationships:

• D to A (horizontal base): 30 inches
• Vertical distance from A to B: 7 inches (probably the first step)
• Vertical distance from B to C: 11 inches (probably the second step)

To find the total height from A to C, you can add the vertical distances:

1. Total vertical height from A to C: \[ \text{Height} = 7 \text{ inches} + 11 \text{ inches} = 18 \text{ inches} \]

Now we have a right triangle with:

• Horizontal leg (BA) = 30 inches
• Vertical leg (AC) = 18 inches

Finding the Length of the Hypotenuse (DC)

Using the Pythagorean Theorem: \[ a^2 + b^2 = c^2 \] where:

• \(a\) is the horizontal leg (30 inches)
• \(b\) is the vertical leg (18 inches)
• \(c\) is the hypotenuse (DC)

Substituting in the lengths we know: \[ 30^2 + 18^2 = c^2 \] Calculating: \[ 900 + 324 = c^2 \] \[ 1224 = c^2 \] \[ c = \sqrt{1224} \approx 34.94 \text{ inches} \]

Summary of Measurements

1. Horizontal leg (DA): 30 inches
2. Vertical height (AB + BC): 18 inches
3. Length of the hypotenuse (DC): Approximately 34.94 inches

If you have specific sections in your diagram labeled for additional measurements, please clarify, and I could assist with those as well. Otherwise, you can use these calculated distances for filling in your geometric diagram.

Answer 10

wait wait sorry The image shows a geometric diagram of a right triangle with points labeled D, A (Base of Stairs), B, and C (Top of Stairs). The distance from D to A is 30 inches, and the vertical distances for the steps are labeled as 7 inches and 11 inches. There are blank spaces for additional measurements along the hypotenuse and the horizontal and vertical sides of the triangle. How long does Emaniee’s ramp need to be?

Answer 11

To determine how long Emaniee's ramp needs to be, we need to find the length of the hypotenuse of the right triangle formed by the ramp, which connects the base of the stairs (point A) to the top of the stairs (point C).

Given Measurements

• Distance from D to A (horizontal distance): 30 inches
• Vertical distances:
  • Height from A to B (first step): 7 inches
  • Height from B to C (top of the stairs): 11 inches

Total Vertical Height

First, calculate the total vertical height from point A (the base) to point C (the top):

\[ \text{Total Height (AC)} = \text{AB} + \text{BC} = 7 \text{ inches} + 11 \text{ inches} = 18 \text{ inches} \]

Right Triangle Dimensions

Now we have the following dimensions for our right triangle:

• Horizontal leg (AD): 30 inches (the base)
• Vertical leg (AC): 18 inches (the total height)

Pythagorean Theorem

Let's apply the Pythagorean theorem to find the hypotenuse (the length of the ramp, \(DC\)):

\[ a^2 + b^2 = c^2 \]

Where:

• \(a = 30\) inches (horizontal distance)
• \(b = 18\) inches (vertical height)
• \(c\) is the hypotenuse (the length of the ramp)

Plugging in the values:

\[ 30^2 + 18^2 = c^2 \]

Calculating:

\[ 900 + 324 = c^2 \] \[ 1224 = c^2 \]

Taking the square root:

\[ c = \sqrt{1224} \approx 34.94 \text{ inches} \]

Conclusion

Emaniee's ramp needs to be approximately 34.94 inches long.