Proportionality>Developing Slope Understanding with Similar Right Triangles(TEKS.Math.8.4.A)
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Texas 8th Grade Math › Proportionality>Developing Slope Understanding with Similar Right Triangles(TEKS.Math.8.4.A)
Points on a line: (2,7), (4,11), (6,15), (8,19)
Using any two points from the line, what is the slope (rate of change) $m$? Use $m=\frac{y_2-y_1}{x_2-x_1}$.
$2$
$\frac{1}{2}$
$-2$
$4$
Explanation
Pick two points, for example $(2,7)$ and $(6,15)$: $m=\frac{15-7}{6-2}=\frac{8}{4}=2$. Using another pair, $(4,11)$ and $(8,19)$: $m=\frac{19-11}{8-4}=\frac{8}{4}=2$. The slope is rise over run (change in $y$ over change in $x$), and it is constant for any two points on the same line. The right triangles you could draw between these point pairs have proportional legs (similar triangles), so the ratio $\frac{\text{rise}}{\text{run}}$ stays $2$.
A line passes through (-3,1) and (5,9).
What is the slope between these two points? Use $m=\frac{y_2-y_1}{x_2-x_1}$.
$8$
$1$
$-1$
$2$
Explanation
Compute rise over run: $m=\frac{9-1}{5-(-3)}=\frac{8}{8}=1$. On this line, moving $8$ units right increases $y$ by $8$ units, so the rate of change is $1$ unit up per $1$ unit right. Any other segment along the same line forms a right triangle similar to this one, so $\frac{\text{rise}}{\text{run}}$ remains $1$.
Points on a line: (1,-2), (4,4), (7,10)
What is the slope between points $(1,-2)$ and $(7,10)$? Use $m=\frac{y_2-y_1}{x_2-x_1}$.
$-2$
$\frac{1}{2}$
$2$
$3$
Explanation
Using $(1,-2)$ and $(7,10)$: $m=\frac{10-(-2)}{7-1}=\frac{12}{6}=2$. Check with another pair, $(1,-2)$ and $(4,4)$: $m=\frac{4-(-2)}{4-1}=\frac{6}{3}=2$. The slope is the ratio of rise to run and is the same for any two points on the line because the right triangles formed by these pairs are similar, keeping $\frac{\text{rise}}{\text{run}}=2$.
A line passes through (0,-5) and (6,-2).
Using any two points, what is the rate of change (slope) of the line? Use $m=\frac{y_2-y_1}{x_2-x_1}$.
$-\frac{1}{2}$
$3$
$-3$
$\frac{1}{2}$
Explanation
Compute rise over run: $m=\frac{-2-(-5)}{6-0}=\frac{3}{6}=\frac{1}{2}$. Any smaller or larger right triangle drawn along the same line (legs parallel to the axes) will be similar, so the ratio $\frac{\text{rise}}{\text{run}}$ remains $\frac{1}{2}$.
Points on a line: (-2,7), (1,1), (4,-5)
Using any two points, what is the slope of the line? Use $m=\frac{y_2-y_1}{x_2-x_1}$.
$2$
$-2$
$\frac{1}{2}$
$-\frac{1}{2}$
Explanation
Using $(-2,7)$ and $(4,-5)$: $m=\frac{-5-7}{4-(-2)}=\frac{-12}{6}=-2$. Check with $(1,1)$ and $(4,-5)$: $m=\frac{-5-1}{4-1}=\frac{-6}{3}=-2$. The negative slope shows $y$ decreases as $x$ increases. Right triangles formed by any two points on this line are similar, so the rise/run ratio is constant at $-2$.