Question

homework
question 23, 1.4.93
part 4 of 10
hw score: 87.82%, 22.83 of 26 points
points: 0.83 of 1
either prove or disprove the statement, “the points (-6, -3), (-2, -4), and (5,1) are the vertices of a right triangle.” use a graph only as a guide.

a. graph, b. graph, c. graph, d. graph

by looking at the graph it seems that the points (-6, -3), (-2, -4), and (5,1) do not form a right triangle.

note that the right triangle has two sides perpendicular to each other. if (-6, -3), (-2, -4), and (5,1) are the vertices of a right triangle, which of the following must be true?
a. two sides of the triangle must have the sum of their slopes as 0.
b. two sides of the triangle must have the same slope.
c. two sides of the triangle must have the product of their slopes as -1.
d. two sides of the triangle must have the product of their slopes as 1.

what is the formula for the slope of a line through \\((x_1,y_1)\\) and \\((x_2,y_2)\\) with \\(x_1 \
eq x_2\\)?
a. \\(\frac{y_2 - y_1}{x_2 - x_1}\\)
b. \\(\frac{y_2 - x_2}{y_1 - x_1}\\)
c. \\(\frac{x_2 - x_1}{y_2 - y_1}\\)
d. \\(\frac{y_1 - x_1}{y_2 - x_2}\\)

show an example get more help - clear all check

Explanation:

First Sub - Question (Multiple - Choice about Right Triangle Slopes)

To determine if a triangle is right - angled, we use the property of perpendicular lines. If two lines are perpendicular, the product of their slopes is - 1.

• Option A: The sum of slopes being 0 is not a property of perpendicular lines. For example, if slope of one line is 1 and another is - 1, their sum is 0 but they are perpendicular (product is - 1), but this is not a general rule for perpendicular lines.
• Option B: Two lines with the same slope are parallel, not perpendicular.
• Option C: This is the correct property of perpendicular lines. If two lines are perpendicular, \(m_1\times m_2=- 1\), where \(m_1\) and \(m_2\) are the slopes of the two lines.
• Option D: The product of slopes being 1 is not a property of perpendicular lines (perpendicular lines have a product of - 1).

The slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) (where \(x_1
eq x_2\)) is calculated as the change in \(y\) - coordinates divided by the change in \(x\) - coordinates. The change in \(y\) is \(y_2 - y_1\) and the change in \(x\) is \(x_2 - x_1\). So the formula for slope is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

• Option A: Matches the formula for slope.
• Option B: \(\frac{y_2 - x_2}{y_1 - x_1}\) is not the formula for slope. It does not represent the change in \(y\) over change in \(x\).
• Option C: \(\frac{x_2 - x_1}{y_2 - y_1}\) is the reciprocal of the slope formula, it represents the change in \(x\) over change in \(y\), which is not the slope.
• Option D: \(\frac{y_1 - x_1}{y_2 - x_2}\) is not related to the slope formula.

Answer:

C. Two sides of the triangle must have the product of their slopes as - 1.

Second Sub - Question (Formula for Slope)