A straight, solid line connects plotted points upper A and upper B on a coordinate plane. The line slopes downward from left to right. Point upper A is left parenthesis negative 3 comma 5 right parenthesis. Point upper B is left parenthesis 7 comma 2 right parenthesis.

Question

To the nearest tenth of a unit, what is the distance between points A and B ?

(1 point)
Responses

10.2 units
10.2 units

10.4 units
10.4 units

9.0 units
9.0 units

10.0 units

Answer 1

To calculate the distance between points A \((-3, 5)\) and B \((7, 2)\) on a coordinate plane, we can use the distance formula:

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

Where \( (x_1, y_1) \) is point A and \( (x_2, y_2) \) is point B. Substituting in the coordinates of points A and B:

\[ (x_1, y_1) = (-3, 5) \quad \text{and} \quad (x_2, y_2) = (7, 2) \]

Now, plug in the values:

\[ d = \sqrt{(7 - (-3))^2 + (2 - 5)^2} \] \[ = \sqrt{(7 + 3)^2 + (2 - 5)^2} \] \[ = \sqrt{(10)^2 + (-3)^2} \] \[ = \sqrt{100 + 9} \] \[ = \sqrt{109} \]

Next, we calculate \(\sqrt{109}\):

Using a calculator: \[ \sqrt{109} \approx 10.4403 \]

Rounding to the nearest tenth gives: \[ \approx 10.4 \]

Therefore, the distance between points A and B, to the nearest tenth of a unit, is 10.4 units.

Answer 2

Suppose a rectangle is drawn on a coordinate plane. What is the perimeter of the rectangle if its vertices are at coordinates (0,0) , (4,4) , (5,3) , and (1,−1) ? Round your answer to the nearest tenth.(1 point)
The perimeter is approximately
units.

Answer 3

To find the perimeter of the rectangle with the vertices at the coordinates \( (0,0) \), \( (4,4) \), \( (5,3) \), and \( (1,-1) \), we first need to calculate the lengths of two adjacent sides.

1. Identifying sides:

   • Let's calculate the distances between consecutive points.

2. Distance between (0, 0) and (4, 4): \[ d_1 = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.656 \]

3. Distance between (4, 4) and (5, 3): \[ d_2 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414 \]

4. Distance between (5, 3) and (1, -1): \[ d_3 = \sqrt{(1 - 5)^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.656 \]

5. Distance between (1, -1) and (0, 0): \[ d_4 = \sqrt{(0 - 1)^2 + (0 + 1)^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414 \]

Now we have the lengths of adjacent sides of the rectangle:

• Length \(L = 4\sqrt{2} \approx 5.656\)
• Width \(W = \sqrt{2} \approx 1.414\)

6. Calculating the perimeter: \[ \text{Perimeter} = 2(L + W) = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) = 10\sqrt{2} \] Using the approximation \(\sqrt{2} \approx 1.414\): \[ \text{Perimeter} \approx 10 \times 1.414 = 14.14 \]

Rounding to the nearest tenth gives the perimeter as approximately: \[ \text{Perimeter} \approx 14.1 , \text{units}. \]

Therefore, the perimeter of the rectangle is approximately 14.1 units.

Answer 4

Suppose a parallelogram is drawn on a coordinate plane. Using vertices with the coordinates (2,1), (2,3), (4,4), and (4,2), compute the perimeter. Round your answer to the nearest tenth.(1 point)
The perimeter is approximately
units.

Answer 5

To compute the perimeter of the parallelogram with vertices at the coordinates \( (2,1) \), \( (2,3) \), \( (4,4) \), and \( (4,2) \), we first identify the lengths of two adjacent sides.

1. Identify the vertices:

   • Let \( A = (2,1) \)
   • Let \( B = (2,3) \)
   • Let \( C = (4,4) \)
   • Let \( D = (4,2) \)

2. Calculate the lengths of two adjacent sides:

   • Length of side AB: \[ AB = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2 \]

   • Length of side AD (or BC): \[ AD = \sqrt{(4 - 2)^2 + (2 - 1)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

3. Calculate the perimeter: The perimeter of a parallelogram is given by: \[ \text{Perimeter} = 2(\text{Length of one side} + \text{Length of the adjacent side}) \] Using \( AB \) (2 units) and \( AD \) (approximately \( \sqrt{5} \)): \[ \text{Perimeter} = 2(AB + AD) = 2(2 + \sqrt{5}) \]

Now, calculate \( \sqrt{5} \) which is approximately \( 2.236 \): \[ \text{Perimeter} \approx 2(2 + 2.236) = 2(4.236) \approx 8.472 \]

4. Round to the nearest tenth: \[ \text{Perimeter} \approx 8.5 \text{ units} \]

Thus, the perimeter of the parallelogram is approximately 8.5 units.

Answer 6

Rectangle upper A upper B upper C upper D is graphed on a coordinate plane. The x-axis ranges from negative 4 to 6 in increments of 1. The y-axis ranges from negative 6 to 2 in increments of 1.
Use the coordinates to compute the approximate area of the rectangle.

(1 point)
Responses

40 units2
40 units squared

10 units2
10 units squared

20 units2
20 units squared

30 units2

Answer 7

To compute the area of rectangle ABCD on a coordinate plane, we need the coordinates of the vertices A, B, C, and D. However, since the specific coordinates have not been given, I'll illustrate how to calculate the area once you have the coordinates.

1. Determine the dimensions of the rectangle:

   • Let's assume the coordinates are \( A(x_1, y_1) \), \( B(x_1, y_2) \), \( C(x_2, y_2) \), and \( D(x_2, y_1) \), where \( x_1 < x_2 \) and \( y_1 < y_2 \).
   • The length of the rectangle is given by the difference in x-coordinates: \[ \text{Length} = x_2 - x_1 \]
   • The width of the rectangle is given by the difference in y-coordinates: \[ \text{Width} = y_2 - y_1 \]

2. Calculate the area:

   • The area \( A \) of the rectangle is calculated by the formula: \[ \text{Area} = \text{Length} \times \text{Width} \]

If we substitute specific values from the coordinate plane into the formula for length and width, we can obtain the exact area.

Example Calculation

For instance, if the rectangle has vertices:

• A at (-4, -6)
• B at (-4, 2)
• C at (6, 2)
• D at (6, -6)

Then:

• Length = \( 6 - (-4) = 6 + 4 = 10 \)
• Width = \( 2 - (-6) = 2 + 6 = 8 \)
• Area \( A = 10 \times 8 = 80 , \text{units}^2 \)

However, based on general options provided:

• 40 units²
• 10 units²
• 20 units²
• 30 units²

It sounds like you might be looking for a specific set of coordinates or contexts to round your answer to those options.

If you have a specific rectangle in mind based on your coordinate plane, please share the vertices, and I can help you calculate the area specific to that rectangle! Otherwise, without the vertex data, the area cannot be determined precisely.

Answer 8

A rectangle is drawn on a coordinate plane with vertices (0,0) , (2,2) , (9,−4) , and (7,−6) . Using the coordinates, what is the approximate area of the rectangle? (1 point)
Responses

24.1 units2
24.1 units squared

52.2 units2
52.2 units squared

26.08 units2
26.07 units squared

12.05 units2

Answer 9

To calculate the area of the rectangle with the given vertices \((0,0)\), \((2,2)\), \((9,-4)\), and \((7,-6)\), we first need to confirm that these four points form a rectangle and then calculate the area.

Step 1: Check the vertices and confirm the rectangle

The coordinates can be categorized as follows:

1. \( A(0,0) \)
2. \( B(2,2) \)
3. \( C(9,-4) \)
4. \( D(7,-6) \)

Step 2: Calculate the lengths of the sides

To find the lengths of the sides, we can calculate the distances between the points:

• Distance \( AB \): \[ d_{AB} = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828 \]

• Distance \( AD \): \[ d_{AD} = \sqrt{(7 - 0)^2 + (-6 - 0)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.219 \]

Step 3: Calculate Area

The area \( A \) of the rectangle is given by the product of its length and width. The two sides we calculated are approximately \( 2.828 \) (width) and \( 9.219 \) (length):

\[ A = AB \times AD \approx (2.828) \times (9.219) \approx 26.05 , \text{units}^2 \]

Step 4: Rounding and choosing the closest option

Rounding \( 26.05 \) to two decimal places gives \( 26.08 , \text{units}^2 \).

Thus, the area of the rectangle is approximately 26.08 units².