Line A passes through the points (6, 1) and (-4, 6). Line B passes through the points (14, 3) and (-10, -9). Which statement is true? Line A overlaps line B. Line A does not inters... Line A passes through the points (6, 1) and (-4, 6). Line B passes through the points (14, 3) and (-10, -9). Which statement is true? Line A overlaps line B. Line A does not intersect line B. Line A intersects line B at exactly one point.
Understand the Problem
The question is asking which statement about the relationship between two lines (A and B) is true based on the given points through which each line passes. The goal is to determine whether the lines overlap, do not intersect, or intersect at exactly one point.
Answer
Line A intersects line B at exactly one point.
Answer for screen readers
Line A intersects line B at exactly one point.
Steps to Solve
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Identify the points for both lines
Line A passes through the points $(6, 1)$ and $(-4, 1)$, so its coordinates are:
- Point A1: $(6, 1)$
- Point A2: $(-4, 1)$
Line B passes through the points $(14, 3)$ and $(-10, -9)$, so its coordinates are:
- Point B1: $(14, 3)$
- Point B2: $(-10, -9)$
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Determine the slope of line A
Since both points of line A have the same $y$-coordinate, the slope is $0$. The equation is:
$$ y = 1 $$ -
Calculate the slope of line B
Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ with point B1 $(14, 3)$ and point B2 $(-10, -9)$:
$$ m_{B} = \frac{-9 - 3}{-10 - 14} = \frac{-12}{-24} = \frac{1}{2} $$The equation of line B in slope-intercept form is:
$$ y = \frac{1}{2}x + b $$
To find $b$, substitute one of the points, say $(14, 3)$:
$$ 3 = \frac{1}{2}(14) + b \implies 3 = 7 + b \implies b = -4 $$
Thus, the equation of line B is:
$$ y = \frac{1}{2}x - 4 $$ -
Analyze the relationship between the lines
- Line A is horizontal ($y = 1$), and Line B has a slope of $\frac{1}{2}$. Since they have different slopes and different $y$-intercepts, they will intersect at exactly one point.
Line A intersects line B at exactly one point.
More Information
Line A is a horizontal line, while line B is not horizontal and has a slope that is not zero, indicating they will meet at one point. This is a typical scenario in analytic geometry where one line intersects another.
Tips
- Assuming same slopes imply overlap: It’s essential to check both the slope and y-intercept. Even if slopes are the same, differing intercepts mean they won't overlap.
- Overlooking horizontal lines: Remember that horizontal lines have a slope of zero, so always take that into account when comparing slopes.
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