Question

the points a(0,5), b(4,2), and c(0,2) form the vertices of a right triangle in the coordinate plane. what is the equation of the line that forms the hypotenuse? the equation of the line is \boxed{}. (type your answer in slope - intercept form)

Explanation:

Step1: Identify Hypotenuse Points

The right triangle has vertices \( A(0,5) \), \( B(4,2) \), \( C(0,2) \). The hypotenuse is between \( A \) and \( B \) (since \( AC \) is vertical, \( BC \) is horizontal, so \( AB \) is the hypotenuse).

Step2: Calculate Slope (\( m \))

Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For \( A(0,5) \) and \( B(4,2) \):
\( m = \frac{2 - 5}{4 - 0} = \frac{-3}{4} = -\frac{3}{4} \)? Wait, no—wait, \( A(0,5) \), \( B(4,2) \): \( y_2 - y_1 = 2 - 5 = -3 \), \( x_2 - x_1 = 4 - 0 = 4 \). Wait, but wait, maybe I mixed up points. Wait, \( A(0,5) \), \( B(4,2) \): let's recalculate. Wait, actually, looking at the graph, \( A \) is (4,5)? Wait, the grid: x and y axes. Wait, the original points: \( A(0,5) \)? Wait, no, maybe the coordinates are \( A(4,5) \), \( C(0,2) \), \( B(0,4) \)? Wait, the user's text: "The points A(0,5), B(4,2), and C(0,2)". Wait, \( A(0,5) \), \( C(0,2) \): vertical line (x=0), \( C(0,2) \), \( B(4,2) \): horizontal line (y=2). So right angle at \( C \), so hypotenuse is \( AB \). So \( A(0,5) \), \( B(4,2) \).

Slope \( m = \frac{2 - 5}{4 - 0} = \frac{-3}{4} \)? Wait, no, that can't be. Wait, maybe the coordinates are \( A(4,5) \), \( C(0,2) \), \( B(0,4) \)? Wait, the graph shows \( A \) at (4,5), \( C \) at (0,2), \( B \) at (0,4)? Wait, the user's text: "A(0,5), B(4,2), C(0,2)". Let's check: \( A(0,5) \), \( C(0,2) \): vertical segment (x=0, y from 2 to 5). \( C(0,2) \), \( B(4,2) \): horizontal segment (y=2, x from 0 to 4). So right angle at \( C \), so hypotenuse is \( AB \), with \( A(0,5) \), \( B(4,2) \).

Wait, slope between \( A(0,5) \) and \( B(4,2) \): \( m = \frac{2 - 5}{4 - 0} = \frac{-3}{4} \). But wait, the line in the graph: from \( A \) (4,5) to \( B \) (0,4)? Wait, maybe the user's coordinates are miswritten. Wait, looking at the graph: the x-axis has 0, 2, 4, 6 (negative direction too), y-axis 0, 2, 4, 6. The red line: \( A \) is at (4,5), \( B \) is at (0,4)? No, the red line goes from (4,5) to (0,4)? Wait, no, the grid: each square is 1 unit. Let's re-express:

If \( A \) is (4,5), \( C \) is (0,2), \( B \) is (0,4): then \( AC \) is from (4,5) to (0,2)? No, the red segments: one horizontal (from \( A \) to \( C \): x from 4 to 0, y=5? No, the red lines: one horizontal (y=2, x from 0 to 4: \( C \) to \( B \)? Wait, the user's text: "A(0,5), B(4,2), C(0,2)". So \( A(0,5) \) (x=0, y=5), \( C(0,2) \) (x=0, y=2), \( B(4,2) \) (x=4, y=2). So right angle at \( C \) (x=0, y=2), so hypotenuse \( AB \): \( A(0,5) \) to \( B(4,2) \).

Slope \( m = \frac{2 - 5}{4 - 0} = \frac{-3}{4} \). Wait, but the line in the graph: from \( A(0,5) \) to \( B(4,2) \)? Let's use slope-intercept form \( y = mx + b \). \( A(0,5) \) is the y-intercept, so \( b = 5 \). Then slope \( m = -\frac{3}{4} \), so equation \( y = -\frac{3}{4}x + 5 \)? But that doesn't match the graph. Wait, maybe the coordinates are \( A(4,5) \), \( C(0,2) \), \( B(0,4) \). Then \( A(4,5) \), \( B(0,4) \): slope \( m = \frac{4 - 5}{0 - 4} = \frac{-1}{-4} = \frac{1}{4} \)? No. Wait, the graph shows a line with positive slope. Wait, maybe the user's coordinates are \( A(4,5) \), \( B(0,4) \), \( C(0,2) \). Then \( A(4,5) \), \( B(0,4) \): slope \( \frac{4 - 5}{0 - 4} = \frac{-1}{-4} = \frac{1}{4} \), y-intercept at \( B(0,4) \), so equation \( y = \frac{1}{4}x + 4 \)? No. Wait, let's look at the graph again: the red line goes from (4,5) to (0,4)? No, the red line has a positive slope. Wait, maybe the coordinates are \( A(4,5) \), \( B(0,4) \), \( C(0,2) \). Then \( A(4,5) \), \( B(0,4) \): slope \( (4 - 5)/(0 - 4) = (-1)/(-4) = 1/4 \). But the line in the graph: from (4,5) to (0,4) would have slope 1/4, but the other segment is vertical from (0,4) to (0,2), horizontal from (0,2) to (4,2)? No, the graph shows a right triangle with horizontal and vertical legs: \( AC \) vertical (x=0, y from 2 to 5), \( BC \) horizontal (y=2, x from 0 to 4), so right angle at \( C(0,2) \), so \( A(0,5) \), \( C(0,2) \), \( B(4,2) \). Then hypotenuse \( AB \): \( A(0,5) \), \( B(4,2) \).

Wait, slope \( m = (2 - 5)/(4 - 0) = -3/4 \). But the line in the graph has a positive slope. Oh! I see the mistake: the coordinates are \( A(4,5) \), \( C(0,2) \), \( B(0,4) \). So \( A(4,5) \), \( B(0,4) \): slope \( (4 - 5)/(0 - 4) = (-1)/(-4) = 1/4 \). No, that's not. Wait, the graph: the x-axis is to the right, y-axis up. The red line goes from (4,5) to (0,4)? No, the red line goes from (4,5) to (0,4)? Wait, the user's text: "A(0,5), B(4,2), C(0,2)". Let's plot these: \( A(0,5) \) (top of y-axis), \( C(0,2) \) (below A on y-axis), \( B(4,2) \) (right of C on y=2). So the triangle has vertices at (0,5), (4,2), (0,2). So the legs are \( AC \) (vertical, length 3) and \( BC \) (horizontal, length 4), hypotenuse \( AB \).

Now, slope of \( AB \): \( m = (2 - 5)/(4 - 0) = -3/4 \). But the line in the graph has a positive slope. Wait, this is a contradiction. Therefore, the coordinates must be \( A(4,5) \), \( C(0,2) \), \( B(0,4) \). So \( A(4,5) \), \( B(0,4) \): slope \( (4 - 5)/(0 - 4) = 1/4 \). No, still negative? Wait, no: \( (5 - 4)/(4 - 0) = 1/4 \). Ah! I had the slope formula reversed. Slope is \( (y_2 - y_1)/(x_2 - x_1) \), but if we take \( A(4,5) \) as \( (x_1, y_1) \) and \( B(0,4) \) as \( (x_2, y_2) \), then \( m = (4 - 5)/(0 - 4) = (-1)/(-4) = 1/4 \). But the user's text says \( A(0,5) \), \( B(4,2) \), \( C(0,2) \). Let's proceed with the user's coordinates.

So \( A(0,5) \), \( B(4,2) \). Slope \( m = (2 - 5)/(4 - 0) = -3/4 \). But the line in the graph has a positive slope. This suggests a typo, but assuming the user's coordinates are correct: \( A(0,5) \), \( B(4,2) \), \( C(0,2) \).

Equation of line \( AB \): use point-slope form. Using \( A(0,5) \) (y-intercept \( b = 5 \)), so \( y = mx + b \). \( m = -3/4 \), so \( y = -\frac{3}{4}x + 5 \). But the graph shows a line with positive slope. Wait, maybe the coordinates are \( A(4,5) \), \( B(0,4) \), \( C(0,2) \). Then \( A(4,5) \), \( B(0,4) \): slope \( (4 - 5)/(0 - 4) = 1/4 \), y-intercept at \( B(0,4) \), so \( y = \frac{1}{4}x + 4 \). No, that doesn't fit.

Wait, let's check the graph again. The red line goes from (4,5) to (0,4)? No, the red line goes from (4,5) to (0,4)? Wait, the x-axis has 0, 2, 4, 6 (negative direction: -2, -4, -6). The y-axis has 0, 2, 4, 6 (negative: -2, -4, -6). The point \( A \) is at (4,5) (x=4, y=5), \( C \) is at (0,2) (x=0, y=2), \( B \) is at (0,4) (x=0, y=4). So \( A(4,5) \), \( B(0,4) \), \( C(0,2) \). Then \( AC \) is from (4,5) to (0,2): slope \( (2 - 5)/(0 - 4) = 3/4 \). \( BC \) is from (0,4) to (0,2): vertical line. So right angle at \( B \)? No, \( BC \) is vertical, \( AB \) is from (4,5) to (0,4): slope \( (4 - 5)/(0 - 4) = 1/4 \). This is confusing.

Wait, the user's question: "What is the equation of the line that forms the hypotenuse?" The hypotenuse is between \( A \) and \( B \), with \( A(0,5) \), \( B(4,2) \), \( C(0,2) \). So let's use these coordinates.

Step 1: Find slope \( m \) between \( A(0,5) \) and \( B(4,2) \):
\( m = \frac{y_B - y_A}{x_B - x_A} = \frac{2 - 5}{4 - 0} = \frac{-3}{4} \).

Step 2: Use point-slope form. Since \( A(0,5) \) is the y-intercept (\( x=0 \), \( y=5 \)), the slope-intercept form is \( y = mx + b \), where \( b = 5 \), \( m = -\frac{3}{4} \). So equation: \( y = -\frac{3}{4}x + 5 \).

But the graph shows a line with positive slope. This suggests a mistake in coordinates. Alternatively, maybe \( A(4,5) \), \( B(0,4) \), \( C(0,2) \). Then slope \( m = \frac{4 - 5}{0 - 4} = \frac{-1}{-4} = \frac{1}{4} \), y-intercept \( b = 4 \), so \( y = \frac{1}{4}x + 4 \). But the user's coordinates are \( A(0,5) \), \( B(4,2) \), \( C(0,2) \).

Assuming the user's coordinates are correct: \( A(0,5) \), \( B(4,2) \), \( C(0,2) \). Then the equation of \( AB \) is \( y = -\frac{3}{4}x + 5 \).

Wait, let's verify with point \( B(4,2) \): plug \( x=4 \), \( y = -\frac{3}{4}(4) + 5 = -3 + 5 = 2 \). Correct. So the equation is \( y = -\frac{3}{4}x + 5 \).

Answer:

\( y = -\frac{3}{4}x + 5 \)