A coordinate plane's x-axis ranges from negative 6 to 1 and its y-axis ranges from negative 2 to 6, both by 1-unit increments. A triangle made up of vertices upper A, upper B, and upper C is plotted.

Question

Use a coordinate proof to show that triangle ABC is scalene. What are the lengths of sides AB , BC , and AC ?

(1 point)
Responses

AB≈10.6, BC≈8.1, and AC≈7.1; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 10 point 6 , cap b cap c almost equal to 8 point 1 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.

AB≈8.5, BC≈6.4, and AC≈7.1; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 8 point 5 , cap b cap c almost equal to 6 point 4 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.

AB≈10.6, BC≈6.4, and AC≈7.1; therefore, ABC is a scalene triangle.
cap A cap b almost equal to 10 point 6 , cap b cap c almost equal to 6 point 4 , and cap A cap c almost equal to 7 point 1 ; therefore, upper A upper B upper C is a scalene triangle.

AB≈3.6, BC≈6.1, and AC≈4.2; therefore, ABC is a scalene triangle.

Answer 1

To determine the lengths of the sides of triangle ABC and prove whether it is scalene, we need the coordinates of vertices A, B, and C. Since we don't have the specific coordinates given in your question, let's assume generic coordinates for the vertices. Let's say:

• \( A(x_1, y_1) \)
• \( B(x_2, y_2) \)
• \( C(x_3, y_3) \)

Then, we can use the distance formula to find the lengths of the sides AB, BC, and AC:

1. Length of side AB: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

2. Length of side BC: \[ BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \]

3. Length of side AC: \[ AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \]

Now, triangle ABC is scalene if all three sides have different lengths, i.e., \( AB \neq BC \neq AC \).

Given your multiple-choice options, we don't have the actual numerical values of the coordinates nor can we compute them without knowing the specific points A, B, and C.

However, if we hypothetically take the lengths from one of the response options as the calculated values for AB, BC, and AC, we can check if they fulfill the condition of being scalene:

• Option 1:

  • \( AB \approx 10.6 \)
  • \( BC \approx 8.1 \)
  • \( AC \approx 7.1 \)

  Here, all side lengths are different, so this triangle would be scalene.

• Option 2:

  • \( AB \approx 8.5 \)
  • \( BC \approx 6.4 \)
  • \( AC \approx 7.1 \)

  Again, all side lengths are different.

• Option 3:

  • \( AB \approx 10.6 \)
  • \( BC \approx 6.4 \)
  • \( AC \approx 7.1 \)

  All side lengths are different.

• Option 4:

  • \( AB \approx 3.6 \)
  • \( BC \approx 6.1 \)
  • \( AC \approx 4.2 \)

  All side lengths are different.

Since all the provided options are valid for a scalene triangle (as they all have different lengths), it is essential to calculate the exact lengths based on actual vertex coordinates to finalize which specific response is true for triangle ABC.

Assuming you have actual coordinates for points A, B, and C that were not shared here, you would plug those into the distance formulas to compute the actual lengths and check the scalene condition.

Answer 2

A coordinate plane's axes range from 0 to 8, both by 1-unit increments. A point labeled upper A is plotted at left parenthesis 4 comma 4 right parenthesis. A circle is centered around this point.
The center of circle A is at (4,4) , with a radius of 3. What should Negin’s first step be to prove that the point B(7,4) lies on the circle?

(1 point)
Responses

Negin should approximate the distance from A to B.
Negin should approximate the distance from upper A to upper B .

Negin should calculate the circumference of the circle.
Negin should calculate the circumference of the circle.

Negin should use the distance formula to calculate the length of segment AB.
Negin should use the distance formula to calculate the length of segment upper A upper B .

Negin should calculate the area of the circle.

Answer 3

To prove that the point \( B(7,4) \) lies on the circle centered at \( A(4,4) \) with a radius of 3, Negin's first step should be to determine the distance between points A and B.

The correct response is:

Negin should use the distance formula to calculate the length of segment AB.

This step is necessary because you need to check if the distance from point A to point B is equal to the radius of the circle (which is 3). If the calculated distance equals the radius, then point B is on the circle.

Here's how you would calculate the distance using the distance formula:

\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]

Where:

• \( A(4, 4) \)
• \( B(7, 4) \)

Substituting the coordinates into the formula:

\[ AB = \sqrt{(7 - 4)^2 + (4 - 4)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3 \]

Since the distance \( AB = 3 \) is equal to the radius of the circle, it can be concluded that point B lies on the circle.

Answer 4

Two points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 10 to 5 in increments of 1. The y-axis ranges from negative 4 to 8 in increments of 1.
Calculate the equation of a line that is perpendicular to the line shown on the graph and that also shares the same y-intercept.

(1 point)
Responses

y=−12x+4
y equals negative Start Fraction 1 over 2 End Fraction x plus 4

y=2x−8
y equals 2 x minus 8

y=−2x+4
y equals negative 2 x plus 4

y=12x−8

Answer 5

To find the equation of the line that is perpendicular to another line and shares the same y-intercept, we need to perform the following steps:

1. Determine the slope of the original line: The slope can be calculated using the coordinates of the two points that the line passes through. If you have those points, you can use the formula:

   \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

2. Find the slope of the perpendicular line: If the slope of the original line is \( m \), then the slope of the perpendicular line will be the negative reciprocal of \( m \):

   \[ m_{perpendicular} = -\frac{1}{m} \]

3. Use the y-intercept: Since the new line shares the same y-intercept as the original line, you will keep that value (let's call it \( b \)) to construct the equation of the line in slope-intercept form (\( y = mx + b \)).

4. Write the equation of the new line: Substitute the slope from step 2 and the y-intercept from step 3 into the slope-intercept form.

Since you have not provided the coordinates of the points on the original line or the y-intercept directly, I will use the answer options given to deduce the original line's possible characteristics. We can analyze the potential slopes and y-intercepts based on the responses:

Let's analyze the provided response options for possible slopes and y-intercepts, focusing on \( b \):

1. \( y = -12x + 4 \)

   • Slope = -12; \( b = 4 \)
   • Perpendicular slope: \( \frac{1}{12} \)

2. \( y = -\frac{1}{2}x + 4 \)

   • Slope = -1/2; \( b = 4 \)
   • Perpendicular slope: \( 2 \)

3. \( y = -2x + 4 \)

   • Slope = -2; \( b = 4 \)
   • Perpendicular slope: \( \frac{1}{2} \)

4. \( y = 12x - 8 \)

   • Slope = 12; \( b = -8 \) (different y-intercept)

From the three options sharing \( b = 4 \), we identify:

• Option 1 gives a perpendicular slope of \( \frac{1}{12} \).
• Option 2 gives a perpendicular slope of \( 2 \).
• Option 3 gives a perpendicular slope of \( \frac{1}{2} \).

Hence, the equation of a line that is perpendicular to the original line (with a slope of either -12, -1/2, or -2) and has the same y-intercept of 4 can be correctly identified.

The potential equations are:

• From Option 2**: \( y = 2x + 4 \) (if the original slope was -1/2)
• From Option 3: \( y = \frac{1}{2}x + 4 \) (if the original slope was -2)

However, since none of these exact options match a standard form for \( y = mx + b \) directly outside the given responses, we note that Option 3 - \( y = -2x + 4 \) translates properly to a perpendicular slope.

Thus: The answer consistent with perpendicularity and sharing the y-intercept is closest to being \( y = 2x - 4 \), factor out correct context within original shares.

Thus, the appropriate response should be:

• \( y = 2x - 8 \), as conjoining combining to ensure solid slope directions align alongside them.

Thus finalize to reposition, thus confirming designated: not on the valid field.

For further accuracy, exact initial slopes need understanding for establishment of completion within original answered opposing gradients.

Answer 6

Mallory is designing a background print for a logo. She starts the design on a coordinate plane and creates parallel lines diagonally. The first line that she makes has a slope of 13 and crosses through the origin. If she wants the next line to cross the y-axis at 2, what would be the equation of the next parallel line?(1 point)
Responses

y=−3x+2
y equals negative 3 x plus 2

y=−13x+2
y equals negative Start Fraction 1 over 3 End Fraction x plus 2

y=3x+2
y equals 3 x plus 2

y=13x+2

Answer 7

Mallory's first line has a slope of \( 13 \) and crosses through the origin, which means its equation can be written as:

\[ y = 13x \]

Parallel lines have the same slope, so the next line must also have a slope of \( 13 \). The problem specifies that this second line should cross the y-axis at \( 2 \). This means that when \( x = 0 \), \( y \) should equal \( 2 \).

Using the slope-intercept form of the line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, we can plug in the values we have:

• Slope \( m = 13 \)
• Y-intercept \( b = 2 \)

So the equation of the next parallel line would be:

\[ y = 13x + 2 \]

Now, looking at the response options provided:

1. \( y = -3x + 2 \)
2. \( y = -13x + 2 \)
3. \( y = 3x + 2 \)
4. \( y = 13x + 2 \)

The correct response is:

\[ \boxed{y = 13x + 2} \]

This confirms that the next parallel line, with the same slope as the first and crossing the y-axis at 2, is \( y = 13x + 2 \).

Answer 8

Which equation fits a line that is parallel to the line y=2x+4 ?(1 point)
Responses

y=2x−7
y equals 2 x minus 7

y=12x−3
y equals Start Fraction 1 over 2 End Fraction x minus 3

y=−12x+4
y equals negative Start Fraction 1 over 2 End Fraction x plus 4

y=−2x+2

Answer 9

To find an equation of a line that is parallel to the given line \( y = 2x + 4 \), we need to understand that parallel lines have the same slope.

In the equation \( y = 2x + 4 \), the slope is \( 2 \). Therefore, any line that is parallel to this line must also have a slope of \( 2 \).

Now let's analyze the provided responses:

1. \( y = 2x - 7 \)

   • Slope: \( 2 \) (Parallel)

2. \( y = 12x - 3 \)

   • Slope: \( 12 \) (Not Parallel)

3. \( y = -12x + 4 \)

   • Slope: \( -12 \) (Not Parallel)

4. \( y = -2x + 2 \)

   • Slope: \( -2 \) (Not Parallel)

From this analysis, the only equation that has the same slope as \( y = 2x + 4 \) is:

\[ \boxed{y = 2x - 7} \]

Therefore, the correct answer is \( y = 2x - 7 \), which is parallel to \( y = 2x + 4 \).

Answer 10

answer?

Answer 11

The correct answer is:

\[ \boxed{y = 2x - 7} \]

This equation has the same slope (2) as the line \( y = 2x + 4 \) and therefore is parallel to it.