Questions: Slope Criteria for Parallel and Perpendicular Lines: Mastery Test ? triangle ABC is a right triangle where m angle B=90°. The coordinates of A and B are (5,0) and (2,5), respectively. If the x-coordinate of point C is 7, what is the y-coordinate of C? 5 7 8 10

Slope Criteria for Parallel and Perpendicular Lines: Mastery Test
? triangle ABC is a right triangle where m angle B=90°. The coordinates of A and B are (5,0) and (2,5), respectively. If the x-coordinate of point C is 7, what is the y-coordinate of C?
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Transcript text: Slope Criteria for Parallel and Perpendicular Lines: Mastery Test ? $\triangle A B C$ is a right triangle where $\mathrm{m} \angle B=90^{\circ}$. The coordinates of $A$ and $B$ are $(5,0)$ and $(2,5)$, respectively. If the $x$-coordinate of point $C$ is 7 , what is the $y$-coordinate of $C$ ? 5 7 8 10

Solution

Solution Steps

To find the y-coordinate of point C in the right triangle $ \triangle ABC $, we can use the fact that the triangle is a right triangle with $ \angle B = 90^\circ $. This means that the lines $ AB $ and $ BC $ are perpendicular. First, calculate the slope of line $ AB $. Then, use the negative reciprocal of this slope to find the slope of line $ BC $. With the x-coordinate of point C given, use the point-slope form of the equation of a line to find the y-coordinate of point C.

Step 1: Calculate the Slope of Line $ AB $

The coordinates of points $ A $ and $ B $ are given as $ A(5, 0) $ and $ B(2, 5) $. The slope $ m_{AB} $ of line $ AB $ is calculated using the formula:

\[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{5 - 0}{2 - 5} = \frac{5}{-3} = -\frac{5}{3} \]

Step 2: Determine the Slope of Line $ BC $

Since $ \triangle ABC $ is a right triangle with $ \angle B = 90^\circ $, the slope $ m_{BC} $ of line $ BC $ is the negative reciprocal of the slope of line $ AB $:

\[ m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{5}{3}} = \frac{3}{5} \]

Step 3: Find the y-coordinate of Point $ C $

Given that the x-coordinate of point $ C $ is $ x_C = 7 $, we can use the point-slope form of the equation of a line to find the y-coordinate $ y_C $:

\[ y - y_B = m_{BC}(x - x_B) \]

Substituting the known values:

\[ y - 5 = \frac{3}{5}(7 - 2) \]

Calculating the right side:

\[ y - 5 = \frac{3}{5} \cdot 5 = 3 \]

Thus, solving for $ y $:

\[ y = 3 + 5 = 8 \]