Question
Step 1: Exploring the Pythagorean Theorem
On the southeast corner of Millennium Park, there is a garden walk. It is marked off in red in the drawing below. Side C, the hypotenuse of the triangle, shows the row along which flowers will be planted.
If side a measures 90 feet and side b measures 120 feet, how many feet of flowers will be planted along side c, the hypotenuse of the triangle? Show your work and explain your reasoning. (2 points)
Page 2 | © Edgenuity Inc.
Calculate the area of the red triangle to find the area of the garden. Show your work. (2 points)
Millennium Park has an outdoor concert theater. Before a concert, the area reserved for special seating is roped off in the shape of a triangle as shown below. How can the converse of the Pythagorean theorem help you determine whether the roped off area is in the shape of a right triangle? (2 points)
Page 3 | © Edgenuity Inc.
The front of the stage, side C, is 50 feet long. A 40-foot rope runs along the side of square B. A 30-foot rope runs along the side of square A. Is the roped off area, triangle ABC, a right triangle? Explain. (2 points)
Page 4 | © Edgenuity Inc.
e) The diagonal of square A, marked off by the red stars, is where the concession stand is located. A local high school band is performing at the outdoor theater on a summer evening. The band has a school banner that is 40-feet long, and band members would like to hang it across the concession stand to let people know they are performing. Estimate the length of the concession stand to determine if the school banner can fit across the length of the concession stand. Show your work and explain your reasoning. (2 points)
Page 5 | © Edgenuity Inc.
Step 2: Finding distance on the coordinate grid
This is a blueprint drawing of the stage area at Millennium Park. The length of one square on the grid is equal to 5 feet. According to the blueprint, the length of the diagonal of square C is 60 feet. Is that correct? Explain. (2 points).
Page 6 | © Edgenuity Inc.
b) If the blueprint is drawn on the coordinate plane with vertices (1, 5) and (11, 15) for the corners labeled with red stars, would that be an accurate representation of the length of the diagonal of the square C? Show your work and explain your reasoning. (4 points—2 points for finding the length of the diagonal; 2 points for explanation)
Page 7 | © Edgenuity Inc.
Step 3: Using the Pythagorean theorem to find the unknown
At another park in Chicago, neighbors get together to plan a community garden. Plans for the community garden are shown below.
a) The members of the gardening group plan to build a walkway through the garden as formed by the hypotenuse of each of the four triangles in the drawing. That way, the gardeners will be able to access all sections of the garden. Calculate the length of the entire walkway to the nearest hundredth of a yard. Show your work. (4 points—1 point for each length calculation and 1 point for the total distance)
Page 8 | © Edgenuity Inc.
Is the value you just wrote for the total length of the walkway a rational or irrational number? Explain. (2 points)
Page 9 | © Edgenuity Inc.
The group is planning to build a fence around the garden. How many yards of fencing materials do they need for the fence? Show your work. (3 points—2 points for finding the value of side b and 1 point for finding the perimeter of the garden)
On the southeast corner of Millennium Park, there is a garden walk. It is marked off in red in the drawing below. Side C, the hypotenuse of the triangle, shows the row along which flowers will be planted.
If side a measures 90 feet and side b measures 120 feet, how many feet of flowers will be planted along side c, the hypotenuse of the triangle? Show your work and explain your reasoning. (2 points)
Page 2 | © Edgenuity Inc.
Calculate the area of the red triangle to find the area of the garden. Show your work. (2 points)
Millennium Park has an outdoor concert theater. Before a concert, the area reserved for special seating is roped off in the shape of a triangle as shown below. How can the converse of the Pythagorean theorem help you determine whether the roped off area is in the shape of a right triangle? (2 points)
Page 3 | © Edgenuity Inc.
The front of the stage, side C, is 50 feet long. A 40-foot rope runs along the side of square B. A 30-foot rope runs along the side of square A. Is the roped off area, triangle ABC, a right triangle? Explain. (2 points)
Page 4 | © Edgenuity Inc.
e) The diagonal of square A, marked off by the red stars, is where the concession stand is located. A local high school band is performing at the outdoor theater on a summer evening. The band has a school banner that is 40-feet long, and band members would like to hang it across the concession stand to let people know they are performing. Estimate the length of the concession stand to determine if the school banner can fit across the length of the concession stand. Show your work and explain your reasoning. (2 points)
Page 5 | © Edgenuity Inc.
Step 2: Finding distance on the coordinate grid
This is a blueprint drawing of the stage area at Millennium Park. The length of one square on the grid is equal to 5 feet. According to the blueprint, the length of the diagonal of square C is 60 feet. Is that correct? Explain. (2 points).
Page 6 | © Edgenuity Inc.
b) If the blueprint is drawn on the coordinate plane with vertices (1, 5) and (11, 15) for the corners labeled with red stars, would that be an accurate representation of the length of the diagonal of the square C? Show your work and explain your reasoning. (4 points—2 points for finding the length of the diagonal; 2 points for explanation)
Page 7 | © Edgenuity Inc.
Step 3: Using the Pythagorean theorem to find the unknown
At another park in Chicago, neighbors get together to plan a community garden. Plans for the community garden are shown below.
a) The members of the gardening group plan to build a walkway through the garden as formed by the hypotenuse of each of the four triangles in the drawing. That way, the gardeners will be able to access all sections of the garden. Calculate the length of the entire walkway to the nearest hundredth of a yard. Show your work. (4 points—1 point for each length calculation and 1 point for the total distance)
Page 8 | © Edgenuity Inc.
Is the value you just wrote for the total length of the walkway a rational or irrational number? Explain. (2 points)
Page 9 | © Edgenuity Inc.
The group is planning to build a fence around the garden. How many yards of fencing materials do they need for the fence? Show your work. (3 points—2 points for finding the value of side b and 1 point for finding the perimeter of the garden)
Answers
GPT-4o mini
### Step 1: Exploring the Pythagorean Theorem
#### Part a: Finding side c (the hypotenuse)
According to the Pythagorean theorem, in a right triangle with sides \( a \), \( b \), and hypotenuse \( c \):
\[
c = \sqrt{a^2 + b^2}
\]
Here, \( a = 90 \) feet and \( b = 120 \) feet.
1. Calculate \( a^2 \) and \( b^2 \):
- \( a^2 = 90^2 = 8100 \)
- \( b^2 = 120^2 = 14400 \)
2. Add \( a^2 \) and \( b^2 \):
\[
a^2 + b^2 = 8100 + 14400 = 22500
\]
3. Take the square root to find \( c \):
\[
c = \sqrt{22500} = 150 \text{ feet}
\]
Thus, the length of the flowers that will be planted along side \( c \) is **150 feet**.
### Part b: Calculating the area of the red triangle
The area \( A \) of a triangle is calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this triangle, we can use \( a \) as the base and \( b \) as the height:
\[
A = \frac{1}{2} \times 90 \times 120
\]
\[
A = \frac{1}{2} \times 10800 = 5400 \text{ square feet}
\]
Thus, the area of the red triangle is **5400 square feet**.
### Part c: Determining if the roped-off area is a right triangle
Using the converse of the Pythagorean theorem, we can determine if triangle ABC is a right triangle by checking if the squares of the lengths of the two shorter sides equal the square of the length of the longest side.
Given:
- Side A = 30 feet
- Side B = 40 feet
- Side C (hypotenuse) = 50 feet
1. Calculate \( 30^2 + 40^2 \):
- \( 30^2 = 900 \)
- \( 40^2 = 1600 \)
- \( 30^2 + 40^2 = 900 + 1600 = 2500 \)
2. Compare with \( 50^2 \):
- \( 50^2 = 2500 \)
Since \( 30^2 + 40^2 = 50^2 \), triangle ABC **is a right triangle**.
### Part d: Estimating the length of the concession stand
The diagonal of square A can be calculated using the Pythagorean theorem:
Given:
- Each side of square A is \( 40 \) feet.
The diagonal \( d \) of a square can be calculated as:
\[
d = s\sqrt{2} = 40\sqrt{2}
\]
Calculating \( 40\sqrt{2} \):
\[
\sqrt{2} \approx 1.414
\]
\[
d \approx 40 \times 1.414 \approx 56.56 \text{ feet}
\]
Therefore, the estimated length of the concession stand is **approximately 56.56 feet**, which means the 40-foot banner **cannot fit across** because it is shorter than the diagonal.
### Step 2: Finding distance on the coordinate grid
#### Part a: Verifying the diagonal length in square C
Each square unit is \( 5 \) feet, so the diagonal of a square with side length \( s \) is:
\[
d = s\sqrt{2}
\]
If the diagonal is 60 feet:
\[
\text{For side length } s, \quad s\sqrt{2} = 60 \implies s = \frac{60}{\sqrt{2}} = 30\sqrt{2} \approx 30 \times 1.414 \approx 42.42 \text{ feet}
\]
Since \( s = 5 \) feet per grid square implies \( \sqrt{(1-11)^2 + (5-15)^2} \) can help determine if these coordinates represent that diagonal.
#### Part b: Calculate the diagonal with vertices (1, 5) and (11, 15)
To find the length (i.e., the diagonal):
\[
\text{Distance} = \sqrt{(11-1)^2 + (15-5)^2} = \sqrt{(10)^2 + (10)^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2}
\]
Calculating \( 10\sqrt{2} \):
\[
10\sqrt{2} \approx 10 \times 1.414 \approx 14.14 \text{ feet}
\]
This is much shorter than the given length of 60 feet, thus **it is not an accurate representation of that diagonal**.
### Step 3: Using the Pythagorean theorem to find the unknown
#### Part a: Length of the entire walkway
Assuming similar triangular calculations, let's assume each of the four triangles holds right triangles, where lengths are provided. Each length will be calculated using pairs \( (base, height) \).
#### Part b: Rational vs irrational number
The square roots derived which aren't whole numbers indicate irrational lengths in decimal.
#### Part c: Fencing requirement
The perimeter (fencing length required) is calculated as adding together all each lengths of border (if they approximate triangular form). It's essential to derive all sides' lengths using the mentioned triangle calculations.
Finally, round-off as necessary while ensuring all calculations for full perimeter represent rational numbers’ context vs rational structure of geometric outlines where decimals are involved based upon community tables presented in the above calculation structures.
---
Feel free to specify when separate fields need more focus, or exact triangle specifications per shape or measurements need distinct presentation.
#### Part a: Finding side c (the hypotenuse)
According to the Pythagorean theorem, in a right triangle with sides \( a \), \( b \), and hypotenuse \( c \):
\[
c = \sqrt{a^2 + b^2}
\]
Here, \( a = 90 \) feet and \( b = 120 \) feet.
1. Calculate \( a^2 \) and \( b^2 \):
- \( a^2 = 90^2 = 8100 \)
- \( b^2 = 120^2 = 14400 \)
2. Add \( a^2 \) and \( b^2 \):
\[
a^2 + b^2 = 8100 + 14400 = 22500
\]
3. Take the square root to find \( c \):
\[
c = \sqrt{22500} = 150 \text{ feet}
\]
Thus, the length of the flowers that will be planted along side \( c \) is **150 feet**.
### Part b: Calculating the area of the red triangle
The area \( A \) of a triangle is calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this triangle, we can use \( a \) as the base and \( b \) as the height:
\[
A = \frac{1}{2} \times 90 \times 120
\]
\[
A = \frac{1}{2} \times 10800 = 5400 \text{ square feet}
\]
Thus, the area of the red triangle is **5400 square feet**.
### Part c: Determining if the roped-off area is a right triangle
Using the converse of the Pythagorean theorem, we can determine if triangle ABC is a right triangle by checking if the squares of the lengths of the two shorter sides equal the square of the length of the longest side.
Given:
- Side A = 30 feet
- Side B = 40 feet
- Side C (hypotenuse) = 50 feet
1. Calculate \( 30^2 + 40^2 \):
- \( 30^2 = 900 \)
- \( 40^2 = 1600 \)
- \( 30^2 + 40^2 = 900 + 1600 = 2500 \)
2. Compare with \( 50^2 \):
- \( 50^2 = 2500 \)
Since \( 30^2 + 40^2 = 50^2 \), triangle ABC **is a right triangle**.
### Part d: Estimating the length of the concession stand
The diagonal of square A can be calculated using the Pythagorean theorem:
Given:
- Each side of square A is \( 40 \) feet.
The diagonal \( d \) of a square can be calculated as:
\[
d = s\sqrt{2} = 40\sqrt{2}
\]
Calculating \( 40\sqrt{2} \):
\[
\sqrt{2} \approx 1.414
\]
\[
d \approx 40 \times 1.414 \approx 56.56 \text{ feet}
\]
Therefore, the estimated length of the concession stand is **approximately 56.56 feet**, which means the 40-foot banner **cannot fit across** because it is shorter than the diagonal.
### Step 2: Finding distance on the coordinate grid
#### Part a: Verifying the diagonal length in square C
Each square unit is \( 5 \) feet, so the diagonal of a square with side length \( s \) is:
\[
d = s\sqrt{2}
\]
If the diagonal is 60 feet:
\[
\text{For side length } s, \quad s\sqrt{2} = 60 \implies s = \frac{60}{\sqrt{2}} = 30\sqrt{2} \approx 30 \times 1.414 \approx 42.42 \text{ feet}
\]
Since \( s = 5 \) feet per grid square implies \( \sqrt{(1-11)^2 + (5-15)^2} \) can help determine if these coordinates represent that diagonal.
#### Part b: Calculate the diagonal with vertices (1, 5) and (11, 15)
To find the length (i.e., the diagonal):
\[
\text{Distance} = \sqrt{(11-1)^2 + (15-5)^2} = \sqrt{(10)^2 + (10)^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2}
\]
Calculating \( 10\sqrt{2} \):
\[
10\sqrt{2} \approx 10 \times 1.414 \approx 14.14 \text{ feet}
\]
This is much shorter than the given length of 60 feet, thus **it is not an accurate representation of that diagonal**.
### Step 3: Using the Pythagorean theorem to find the unknown
#### Part a: Length of the entire walkway
Assuming similar triangular calculations, let's assume each of the four triangles holds right triangles, where lengths are provided. Each length will be calculated using pairs \( (base, height) \).
#### Part b: Rational vs irrational number
The square roots derived which aren't whole numbers indicate irrational lengths in decimal.
#### Part c: Fencing requirement
The perimeter (fencing length required) is calculated as adding together all each lengths of border (if they approximate triangular form). It's essential to derive all sides' lengths using the mentioned triangle calculations.
Finally, round-off as necessary while ensuring all calculations for full perimeter represent rational numbers’ context vs rational structure of geometric outlines where decimals are involved based upon community tables presented in the above calculation structures.
---
Feel free to specify when separate fields need more focus, or exact triangle specifications per shape or measurements need distinct presentation.