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

7 answers

To determine whether triangle ABC is scalene using a coordinate proof, we first need to assign coordinates to the points A, B, and C. Once we have the coordinates, we can find the lengths of the sides using the distance formula.

Let’s assign the following coordinates:

  • Point A (0, 0)
  • Point B (10, 6)
  • Point C (7, 1)

Now, we will calculate the lengths of sides AB, BC, and AC:

Using the Distance Formula: The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

  1. Calculating AB: \[ AB = \sqrt{(10 - 0)^2 + (6 - 0)^2} = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.7 \]

  2. Calculating BC: \[ BC = \sqrt{(7 - 10)^2 + (1 - 6)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83 \]

  3. Calculating AC: \[ AC = \sqrt{(7 - 0)^2 + (1 - 0)^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} \approx 7.07 \]

Now we have the approximate lengths of the sides:

  • \(AB \approx 11.7\)
  • \(BC \approx 5.83\)
  • \(AC \approx 7.07\)

Since all three sides have different lengths, triangle ABC is a scalene triangle.

Conclusion: The lengths of the sides are \( AB \approx 11.7\), \(BC \approx 5.83\), and \(AC \approx 7.07\); therefore, triangle ABC is a scalene triangle.

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

To prove that the point \( B(7, 4) \) lies on the circle with center \( A(4, 4) \) and a radius of \( 3 \), Negin's first step should be to use the distance formula to calculate the length of segment AB.

Here’s why this is the appropriate choice:

  1. The equation for a point to lie on a circle is determined by its distance from the center being equal to the radius.
  2. The distance between points \( A(4, 4) \) and \( B(7, 4) \) can be calculated using the distance formula.
  3. If the calculated distance equals the radius of the circle (which is 3), then point B is indeed on the circle.

So, the correct response is: Negin should use the distance formula to calculate the length of segment AB.

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=−2x+4 y equals negative 2 x plus 4 y=12x−8 y equals Start Fraction 1 over 2 End Fraction x minus 8 y=2x−8 y equals 2 x minus 8 y=−12x+4 y equals negative Start Fraction 1 over 2 End Fraction x plus 4

To calculate the equation of a line that is perpendicular to a given line and shares the same y-intercept, we need to follow these steps:

  1. Identify the slope of the original line: The slope of a line is typically written in the form \(y = mx + b\), where \(m\) is the slope. If you have or can describe the original line's equation, you can find this slope.

  2. Find the slope of the perpendicular line: The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. If the original slope is \(m\), the perpendicular slope will be \(-1/m\).

  3. Use the same y-intercept: The y-intercept \(b\) will be the same as that of the original line.

Since the specific slope of the original line is not provided, let's assume the slope of the original line is either \(m = 2\) or \(m = -\frac{1}{2}\) based on the provided responses.

Typical Scenario:

  • If the original line's slope is \(2\):

    • The perpendicular slope would be \(-\frac{1}{2}\).
    • If the y-intercept is \(b = 4\), then the equation of the perpendicular line would be: \[ y = -\frac{1}{2}x + 4 \]
  • If the original line's slope is \(-\frac{1}{2}\):

    • The perpendicular slope would be \(2\).
    • If the y-intercept is \(b = 4\), then the equation of the perpendicular line would be: \[ y = 2x + 4 \]
  • If the original line's slope is \(-2\):

    • The perpendicular slope would be \(\frac{1}{2}\).
    • If the y-intercept is \(b = 4\), then the equation of the perpendicular line would be: \[ y = \frac{1}{2}x + 4 \]

Without further context about the original line, I can help you choose from the given options assuming a few things:

  • If we take \(b = 4\) and the original slope is either \(2\) or \(-2\), the resulting equations for the responses are as follows:

Evaluating Responses:

  • \(y = -2x + 4\) (this is not perpendicular if the original line has a slope of \(2\), rather it is parallel).
  • \(y = \frac{1}{2}x + 4\) (perpendicular if original line has a slope of \(2\)).
  • \(y = 12x - 8\) (not related).
  • \(y = 2x - 8\) (parallel with a different intercept).
  • \(y = -12x + 4\) (not related).
  • \(y = -\frac{1}{2}x + 4\) (this is also a possible scenario for perpendicular).

Conclusion:

The most plausible response for a line perpendicular to a line with slope \(2\) and same y-intercept would be:

  • \(y = -\frac{1}{2}x + 4\) (if the slope of the original is \(2\))

Please select the option that best fits the context of the original line provided in your problem.

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

To find the equation of the second line that is parallel to the first line, we first need to understand the properties of parallel lines:

  1. Parallel Lines Have the Same Slope: Since the first line has a slope of \(13\), any line parallel to it will also have a slope of \(13\).

  2. Finding the Equation: The standard form of a line's equation is given by the slope-intercept form: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept.

Since Mallory wants the next line to have a y-intercept of \(2\), we can plug in the slope and the y-intercept into the equation of the line:

  • Slope (m): \(13\)
  • Y-Intercept (b): \(2\)

Now we can write the equation of the next line:

\[ y = 13x + 2 \]

Thus, the equation of the next parallel line is: \(y = 13x + 2\).

This option is available in your responses:

  • y = 13x + 2

So, the correct answer is: \(y = 13x + 2\).