Slopes of Parallel and Perpendicular Lines

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Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 25 · 1 point

Worked example

Write the line perpendicular to $y=\frac{2}{3}x-1$ that passes through $(2,5)$.

1. The original slope is $\frac{2}{3}$, so the perpendicular slope is $-\frac{3}{2}$.
2. Use point-slope form: $y-5=-\frac{3}{2}(x-2)$.
3. Simplify: $y=-\frac{3}{2}x+3+5$.

$y=-\frac{3}{2}x+8$

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TEKSForge practice · unscored

1. 1. What slope is perpendicular to the line $3x+y=7$?

   Show answer and reasoning

   $\frac{1}{3}$

   Rewrite as $y=-3x+7$. The negative reciprocal of −3 is $\frac{1}{3}$.

2. 2. Are $y=2x+1$ and $y=-0.5x+4$ perpendicular?

   Show answer and reasoning

   Yes

   The slopes are 2 and −0.5, and $2(-0.5)=-1$.

3. 3. Write a line parallel to the line through (2, 7) and (6, 1) with y-intercept (0, −4).

   Show answer and reasoning

   $y=-\frac{3}{2}x-4$

   The original slope is $\frac{1-7}{6-2}=-\frac{3}{2}$. Keep that slope and use $b=-4$.

4. 4. Quadrilateral ABCD has vertices A(0, 0), B(4, 2), C(3, 4), and D(−1, 2). Use slopes to decide whether it is a rectangle.

   Show answer and reasoning

   Yes, it is a rectangle.

   $m_{AB}=m_{CD}=\frac{1}{2}$ and $m_{BC}=m_{DA}=-2$, so both opposite-side pairs are parallel. Adjacent slopes multiply to −1, so the parallelogram has right angles and is a rectangle.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 14 · 1 point

The equation of a line containing one leg of a right triangle is $y = -4x$. Which equation could represent the line containing the other leg of this triangle?

1. $y = -\frac{1}{4}x$
2. $y = \frac{1}{4}x + 2$
3. $y = 4x$
4. $y = -4x + 2$

Show STAAR answer and explanation

$y = \frac{1}{4}x + 2$

Perpendicular nonvertical lines have opposite reciprocal slopes. The negative reciprocal of −4 is $1/4$, and the y-intercept 2 does not affect perpendicularity.

$y = -\frac{1}{4}x$: This is the reciprocal but not the opposite sign; perpendicular slopes must multiply to −1.

$y = 4x$: The sign changed, but 4 is not the reciprocal of −4; the slope must be $1/4$.

$y = -4x + 2$: This line is parallel to the given line because both slopes are −4, not perpendicular.

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Archived released STAAR Geometry EOC · 2013 · Item 25 · 1 point

The graph of line $g$ is shown. Which equation describes a line parallel to line $g$ that has a y-intercept at $(0, -1)$?

1. $y = 2x - 1$
2. $y = \frac{1}{2}x - 1$
3. $y = -\frac{1}{2}x - 1$
4. $y = -2x - 1$

Show STAAR answer and explanation

$y = -\frac{1}{2}x - 1$

The slope of g through (−4, 6) and (4, 2) is $(2-6)/(4-(-4))=-4/8=-1/2$. A parallel line has the same slope; with intercept −1 its equation is $y=-\frac12x-1$.

$y = 2x - 1$: Slope 2 is the negative reciprocal, so this line is perpendicular rather than parallel.

$y = \frac{1}{2}x - 1$: This has the wrong slope sign; g decreases from left to right.

$y = -2x - 1$: This uses the reciprocal slope −2 instead of −1/2.

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Archived released STAAR Geometry EOC · 2013 · Item 47 · 1 point

The slopes of the sides of quadrilateral $ABCD$ are shown in the table. Which statement describes the relationships between the sides of the quadrilateral?

Slopes of the Sides of Quadrilateral ABCD

Side	Slope
$\overline{AB}$	$\frac{2}{5}$
$\overline{BC}$	$-\frac{2}{5}$
$\overline{CD}$	$\frac{2}{5}$
$\overline{AD}$	$-\frac{5}{2}$

1. $\overline{AD}$ is parallel to $\overline{BC}$, but $\overline{AB}$ is not parallel to $\overline{CD}$.
2. $\overline{AB}$ is parallel to $\overline{CD}$, but $\overline{AD}$ is not parallel to $\overline{BC}$.
3. $\overline{AB}$ is parallel to $\overline{CD}$, and $\overline{AD}$ is parallel to $\overline{BC}$.
4. $\overline{AD}$ is not parallel to $\overline{BC}$, and $\overline{AB}$ is not parallel to $\overline{CD}$.

Show STAAR answer and explanation

$\overline{AB}$ is parallel to $\overline{CD}$, but $\overline{AD}$ is not parallel to $\overline{BC}$.

AB and CD both have slope 2/5, so they are parallel. AD has slope −5/2 while BC has slope −2/5; unequal slopes mean those sides are not parallel.

$\overline{AD}$ is parallel to $\overline{BC}$, but $\overline{AB}$ is not parallel to $\overline{CD}$.: This reverses both comparisons: AB and CD have equal slopes; AD and BC do not.

$\overline{AB}$ is parallel to $\overline{CD}$, and $\overline{AD}$ is parallel to $\overline{BC}$.: AD and BC have different negative slopes; sharing a sign does not make lines parallel.

$\overline{AD}$ is not parallel to $\overline{BC}$, and $\overline{AB}$ is not parallel to $\overline{CD}$.: AB and CD have the same slope, so that pair is parallel.

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