Question

multiple choice 1 point igor believes the slope of ac is greater than the slope of df, while keenan believes the two slopes are equal. who do you agree with? igor. the slope of ac is greater than the slope of df. keenan. the slopes of similar triangles are equal. clear my selection

Explanation:

Step1: Find coordinates of points for AC

From the graph, let's identify points for line \( AC \). Let's assume point \( A \) is at \( (7, -5) \) and point \( C \) is at \( (-3, 3) \) (by counting grid squares). The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). So for \( AC \), \( m_{AC} = \frac{3 - (-5)}{-3 - 7} = \frac{8}{-10} = -\frac{4}{5} \).

Step2: Find coordinates of points for DF

For line \( DF \), let's find points \( D \) and \( F \). From the graph, \( D \) seems to be at \( (4, -1) \) and \( F \) at \( (-2, 3) \). Using slope formula, \( m_{DF} = \frac{3 - (-1)}{-2 - 4} = \frac{4}{-6} = -\frac{2}{3} \)? Wait, no, maybe I misread the points. Wait, maybe better to use the right triangles. The triangle for \( AC \): vertical change and horizontal change. From \( B \) to \( C \) and \( B \) to \( A \). Wait, \( B \) is at \( (-3, -5) \), \( C \) at \( (-3, 3) \), so vertical change \( 3 - (-5) = 8 \), horizontal change \( 7 - (-3) = 10 \), slope \( -8/10 = -4/5 \). For \( DF \): \( E \) is at \( (-2, -1) \), \( F \) at \( (-2, 3) \), \( D \) at \( (4, -1) \). So vertical change from \( F \) to \( D \): \( -1 - 3 = -4 \), horizontal change \( 4 - (-2) = 6 \), slope \( -4/6 = -2/3 \)? Wait, no, maybe the triangles are similar. Wait, actually, lines \( AC \) and \( DF \) are part of the same line? Wait, no, the graph shows that \( AC \) and \( DF \) are parallel? Wait, no, the key is that similar triangles have equal slopes because slope is rise over run, and similar triangles have proportional rise and run, so the ratio (slope) is equal. Wait, maybe I made a mistake in coordinates. Let's re-express: For line \( AC \), let's take two points. Let's say \( A \) is \( (7, -5) \), \( C \) is \( (-3, 3) \). The run (horizontal change) is \( -3 - 7 = -10 \), rise (vertical change) is \( 3 - (-5) = 8 \), so slope \( 8 / (-10) = -4/5 \). For line \( DF \), let's take \( D(4, -1) \) and \( F(-2, 3) \). Run: \( -2 - 4 = -6 \), rise: \( 3 - (-1) = 4 \), slope \( 4 / (-6) = -2/3 \). Wait, that's not equal. Wait, no, maybe the triangles are similar, so their slopes should be equal. Wait, maybe I picked the wrong points. Wait, the triangle for \( AC \): from \( B(-3, -5) \) to \( C(-3, 3) \) (vertical) and \( B(-3, -5) \) to \( A(7, -5) \) (horizontal). So rise is \( 8 \), run is \( 10 \), slope \( -8/10 = -4/5 \). For \( DF \): from \( E(-2, -1) \) to \( F(-2, 3) \) (vertical) and \( E(-2, -1) \) to \( D(4, -1) \) (horizontal). Rise is \( 4 \), run is \( 6 \), slope \( -4/6 = -2/3 \). Wait, but \( 8/10 = 4/5 \), \( 4/6 = 2/3 \)? No, that's not equal. Wait, maybe the lines are the same? Wait, no, the graph shows that \( AC \) and \( DF \) are segments of the same line? Wait, let's check if \( D \) and \( F \) lie on \( AC \). Let's plug \( x = 4 \) into the equation of \( AC \). The equation of \( AC \): using point \( A(7, -5) \) and slope \( -4/5 \), \( y - (-5) = -4/5(x - 7) \), \( y + 5 = -4/5 x + 28/5 \), \( y = -4/5 x + 28/5 - 25/5 = -4/5 x + 3/5 \). At \( x = 4 \), \( y = -16/5 + 3/5 = -13/5 = -2.6 \), but \( D \) is at \( (4, -1) \), which is not on that line. Wait, maybe I misidentified the points. Alternatively, the triangles \( ABC \) and \( DEF \) are similar, so their corresponding sides are proportional, meaning the slope (rise over run) is the same. Because slope is rise/run, and for similar triangles, rise1/run1 = rise2/run2 (since they are proportional), so the slopes are equal. Therefore, Keenan is correct because similar triangles have equal slopes (same rise over run ratio).

Answer:

Keenan. The slopes of similar triangles are equal.