Parallel & Perpendicular Lines - Math 1

Line $\ell$ is perpendicular to the line with equation $2x + 5y = 15$ and passes through the point $(4, -1)$ . What is the equation of line $\ell$ in slope-intercept form?
Answer: $y = \frac{5}{2}x - 11$ When you see a problem about perpendicular lines, remember that perpendicular lines have slopes that are negative reciprocals of each other. If one line has slope $m$ , a perpendicular line has slope $-\frac{1}{m}$ . First, find the slope of the given line $2x + 5y = 15$ . Convert to slope-intercept form by solving for $y$ : $5y = -2x + 15$ $y = -\frac{2}{5}x + 3$ The slope is $-\frac{2}{5}$ , so a perpendicular line must have slope $-\frac{1}{(-\frac{2}{5})} = \frac{5}{2}$ . Now use point-slope form with the point $(4, -1)$ and slope $\frac{5}{2}$ : $y - (-1) = \frac{5}{2}(x - 4)$ $y + 1 = \frac{5}{2}x - 10$ $y = \frac{5}{2}x - 11$ This matches answer choice D. Looking at the wrong answers: A has the wrong slope ( $-\frac{5}{2}$ ) - this would be parallel to a line with slope $-\frac{5}{2}$ , not perpendicular to our given line. B uses the same slope as the original line ( $-\frac{2}{5}$ ), making it parallel rather than perpendicular. C has the correct slope but the wrong y-intercept - this represents an error in the algebra when solving for the y-intercept. Study tip: Always double-check that perpendicular slopes multiply to give $-1$ . Here: $\frac{5}{2} \times (-\frac{2}{5}) = -1$ ✓
Two lines have equations $y = kx + 3$ and $x + 4y = 12$ . If these lines are parallel, what is the value of $k$ ?
Answer: $k = -\frac{1}{4}$ When you encounter parallel lines, remember that parallel lines have identical slopes but different y-intercepts. Your task is to find the slopes of both lines and set them equal. The first line $y = kx + 3$ is already in slope-intercept form ( $y = mx + b$ ), so its slope is $k$ . For the second line $x + 4y = 12$ , you need to solve for $y$ to find the slope. Subtract $x$ from both sides: $4y = -x + 12$ . Then divide everything by 4: $y = -\frac{1}{4}x + 3$ . The slope is $-\frac{1}{4}$ . Since parallel lines have equal slopes, $k = -\frac{1}{4}$ . Looking at the wrong answers: Choice A gives $k = -4$ , which is the negative reciprocal of the correct slope—this would make the lines perpendicular, not parallel. Choice B gives $k = \frac{1}{4}$ , which is the opposite sign of the correct answer; this is a common error when manipulating negative fractions. Choice C gives $k = 4$ , which might come from incorrectly thinking the slope of $x + 4y = 12$ is the coefficient of $x$ (1) times the coefficient of $y$ (4). Study tip: Always convert both equations to slope-intercept form when working with parallel or perpendicular lines. Remember that parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.
Line $M$ passes through $(3, -1)$ and has a slope that is twice the slope of the line $2x - 3y = 9$ . Line $N$ is perpendicular to line $M$ and passes through the origin. What is the equation of line $N$ ?
Answer: $y = -\frac{3}{4}x$ This problem tests your understanding of slope relationships and perpendicular lines. When you see questions involving "twice the slope" or "perpendicular," you need to work systematically through the slope relationships. First, find the slope of the given line $2x - 3y = 9$ . Convert to slope-intercept form: $-3y = -2x + 9$ , so $y = \frac{2}{3}x - 3$ . The slope is $\frac{2}{3}$ . Line $M$ has a slope that's twice this: $2 \cdot \frac{2}{3} = \frac{4}{3}$ . Since line $N$ is perpendicular to line $M$ , its slope is the negative reciprocal of $\frac{4}{3}$ , which is $-\frac{3}{4}$ . Line $N$ passes through the origin $(0,0)$ , so its equation is $y = -\frac{3}{4}x$ . Choice A ( $y = \frac{4}{3}x$ ) gives you the slope of line $M$ , not its perpendicular. This is the trap if you forget that perpendicular lines have negative reciprocal slopes. Choice C ( $y = -\frac{4}{3}x$ ) takes the negative of line $M$ 's slope but forgets the reciprocal part of "negative reciprocal." Choice D ( $y = \frac{3}{4}x$ ) gives you the reciprocal but forgets the negative, which would make it perpendicular to a line with slope $-\frac{4}{3}$ , not $\frac{4}{3}$ . The answer is B. Strategy tip: For perpendicular lines, always remember both parts: negative AND reciprocal. If the original slope is $\frac{a}{b}$ , the perpendicular slope is $-\frac{b}{a}$ .
Two parallel lines have equations $L_1: ax + by = c$ and $L_2: dx + ey = f$ , where all coefficients are non-zero. Which condition must be satisfied for these lines to be parallel but not identical?
Answer: $\frac{a}{d} = \frac{b}{e} \neq \frac{c}{f}$ and the slope ratios are equal but intercepts differ For lines $ax + by = c$ and $dx + ey = f$ to be parallel, their slopes must be equal. Converting to slope-intercept form gives slopes $-\frac{a}{b}$ and $-\frac{d}{e}$ . Setting these equal: $-\frac{a}{b} = -\frac{d}{e} \Rightarrow \frac{a}{d} = \frac{b}{e}$ . For the lines to be distinct (not identical), they cannot have the same y-intercept, which means $\frac{c}{f} \neq \frac{a}{d}$ . Choice A describes identical lines. Choice C uses determinant notation but is equivalent to choice B. Choice D incorrectly describes perpendicular lines.
Line $L_1$ has equation $y = mx + 3$ where $m > 0$ . Line $L_2$ passes through points $(2, 7)$ and $(8, 1)$ . For what value of $m$ will $L_1$ be perpendicular to $L_2$ ?
Answer: $m = 1$ First, find the slope of L₂ using the two given points: slope = (1-7)/(8-2) = -6/6 = -1. For L₁ to be perpendicular to L₂, their slopes must be negative reciprocals. Since L₂ has slope -1, L₁ must have slope that satisfies: m × (-1) = -1, so m = 1. We can verify: m = 1 and the constraint m > 0 is satisfied.
Line $\alpha$ passes through points $(2, -3)$ and $(8, 1)$ , while line $\beta$ has equation $2x - 3y = 9$ . A third line $\gamma$ is perpendicular to line $\alpha$ and parallel to a line with slope $-\frac{3}{2}$ . What is the slope of line $\gamma$ ?
Answer: $m_{\gamma} = -\frac{3}{2}$ First, find the slope of line α: $m_{\alpha} = \frac{1-(-3)}{8-2} = \frac{4}{6} = \frac{2}{3}$ . Since line γ is perpendicular to line α, its slope is the negative reciprocal: $m_{\gamma} = -\frac{3}{2}$ . The problem also states that γ is parallel to a line with slope $-\frac{3}{2}$ , which confirms our answer since parallel lines have equal slopes. Choice A gives the slope of line α, not its perpendicular. Choice C gives the positive reciprocal instead of negative reciprocal. Choice D gives the negative reciprocal of the wrong value.
A rectangle has vertices at $A(1, 2)$ , $B(7, 4)$ , $C(6, 7)$ , and $D(0, 5)$ . To verify this is indeed a rectangle, which slopes should be calculated and what should be true about their relationships?
Answer: Calculate slopes of all four sides; opposite sides should have equal slopes and adjacent sides should have slopes that are negative reciprocals When verifying that a quadrilateral is a rectangle, you need to confirm two key properties: opposite sides are parallel (equal slopes) and adjacent sides are perpendicular (slopes are negative reciprocals). This requires examining all four sides systematically. To verify this rectangle, calculate the slopes of all four sides: $AB = \frac{4-2}{7-1} = \frac{1}{3}$ , $BC = \frac{7-4}{6-7} = -3$ , $CD = \frac{5-7}{0-6} = \frac{1}{3}$ , and $DA = \frac{2-5}{1-0} = -3$ . Notice that opposite sides $AB$ and $CD$ both have slope $\frac{1}{3}$ , while opposite sides $BC$ and $DA$ both have slope $-3$ . Additionally, adjacent sides have slopes that are negative reciprocals: $\frac{1}{3} \times (-3) = -1$ . This confirms the shape is indeed a rectangle. Choice A is wrong because having all different slopes would mean no sides are parallel, which contradicts the definition of a rectangle. Choice B is incorrect because checking only two adjacent sides doesn't verify that the opposite pair of sides is also parallel—you need to confirm the parallel property for both pairs of opposite sides. Choice C is flawed because diagonal slopes being negative reciprocals doesn't guarantee right angles at the vertices; this approach doesn't verify the essential parallel and perpendicular relationships of the sides themselves. Remember: for rectangle verification problems, always check that opposite sides are parallel (equal slopes) AND adjacent sides are perpendicular (negative reciprocal slopes). This requires calculating all four side slopes.
Consider the family of lines $y = mx + b$ where $m$ and $b$ are parameters. Line $L$ from this family passes through $(3, 7)$ and is perpendicular to the line $4x - y = 8$ . What is the equation of line $L$ ?
Answer: $y = -\frac{1}{4}x + \frac{31}{4}$ When you encounter problems about perpendicular lines, remember that perpendicular lines have slopes that are negative reciprocals of each other. This means if one line has slope $m$ , a perpendicular line has slope $-\frac{1}{m}$ . First, find the slope of the given line $4x - y = 8$ . Rewrite this in slope-intercept form: $y = 4x - 8$ . The slope is 4, so any line perpendicular to this must have slope $-\frac{1}{4}$ . Now use the point-slope form with the point $(3, 7)$ and slope $-\frac{1}{4}$ : $y - 7 = -\frac{1}{4}(x - 3)$ $y - 7 = -\frac{1}{4}x + \frac{3}{4}$ $y = -\frac{1}{4}x + \frac{3}{4} + 7$ $y = -\frac{1}{4}x + \frac{3}{4} + \frac{28}{4}$ $y = -\frac{1}{4}x + \frac{31}{4}$ This matches answer choice C. Answer A has the correct slope but wrong y-intercept ( $\frac{25}{4}$ instead of $\frac{31}{4}$ ) - this likely comes from arithmetic errors when combining fractions. Answer B uses slope 4, which would make the line parallel to the given line, not perpendicular. Answer D uses positive slope $\frac{1}{4}$ , missing the crucial negative sign needed for perpendicularity. Study tip: Always double-check perpendicular slope calculations by verifying that the product of the two slopes equals -1. Here: $4 \times (-\frac{1}{4}) = -1$ ✓
Line $j$ passes through points $(-2, 5)$ and $(4, -1)$ . Line $k$ has the equation $y = mx + 7$ . For what value of $m$ will lines $j$ and $k$ be perpendicular?
Answer: $m = 1$ First, find the slope of line j using the slope formula: $m_j = \frac{-1-5}{4-(-2)} = \frac{-6}{6} = -1$ . For perpendicular lines, their slopes are negative reciprocals. Since the slope of line j is -1, the slope of line k must be $-\frac{1}{-1} = 1$ . Choice B gives the slope of a parallel line. Choice C gives the reciprocal without the negative sign. Choice D gives the negative reciprocal of the wrong slope.
A rectangle has vertices at $A(1, 2)$ , $B(7, 4)$ , $C(6, 7)$ , and $D(0, 5)$ . To verify this is actually a rectangle, which slopes should be calculated and what relationship should they satisfy?
Answer: Calculate slopes of all four sides; opposite sides should be equal and adjacent sides should be negative reciprocals When verifying that a quadrilateral is a rectangle, you need to confirm two key properties: opposite sides are parallel (same slopes) and adjacent sides are perpendicular (slopes are negative reciprocals with product -1). To properly verify this shape is a rectangle, you must calculate all four side slopes. Let's see why: The slopes are $m_{AB} = \frac{4-2}{7-1} = \frac{1}{3}$ , $m_{BC} = \frac{7-4}{6-7} = -3$ , $m_{CD} = \frac{5-7}{0-6} = \frac{1}{3}$ , and $m_{DA} = \frac{2-5}{1-0} = -3$ . Notice that opposite sides AB and CD both have slope $\frac{1}{3}$ , while opposite sides BC and DA both have slope -3, confirming parallel opposite sides. Additionally, adjacent sides have slopes $\frac{1}{3}$ and -3, which are negative reciprocals since $\frac{1}{3} \times (-3) = -1$ , confirming perpendicular adjacent sides. Answer A is incomplete because checking only two adjacent sides doesn't verify that the opposite sides are parallel. Answer B focuses on diagonals, but diagonal slopes don't determine whether a quadrilateral is a rectangle—they could be negative reciprocals in other quadrilaterals too. Answer D is completely wrong since having all different slopes would actually prevent the shape from being a rectangle. Remember: To verify any rectangle, always check all four sides. You need both conditions satisfied: opposite sides parallel (equal slopes) and adjacent sides perpendicular (negative reciprocal slopes).
In a coordinate plane, line $\ell_1$ has equation $y - 4 = \frac{2}{5}(x + 3)$ and line $\ell_2$ passes through points $(1, -2)$ and $(-4, 3)$ . A third line $\ell_3$ is drawn such that it is parallel to $\ell_1$ and perpendicular to $\ell_2$ . What must be true about the slope of $\ell_3$ ?
Answer: No such line $\ell_3$ exists because the conditions create contradictory slope requirements First, identify the slope of line ℓ₁. The equation y - 4 = (2/5)(x + 3) is in point-slope form, so the slope is 2/5. Next, find the slope of line ℓ₂ using points (1, -2) and (-4, 3): slope = (3 - (-2))/(-4 - 1) = 5/(-5) = -1. For line ℓ₃ to be parallel to ℓ₁, it must have the same slope as ℓ₁, which is 2/5. For line ℓ₃ to be perpendicular to ℓ₂, its slope must be the negative reciprocal of -1, which is -1/(-1) = 1. However, line ℓ₃ cannot simultaneously have slope 2/5 (to be parallel to ℓ₁) and slope 1 (to be perpendicular to ℓ₂). Since 2/5 ≠ 1, no such line exists that satisfies both conditions.
A quadrilateral $PQRS$ has vertices $P(-1, 2)$ , $Q(3, 4)$ , $R(5, 0)$ , and $S(1, -2)$ . Which pair of opposite sides are parallel?
Answer: Sides $PQ$ and $RS$ are parallel because both have slope $\frac{1}{2}$ Calculate all slopes: $m_{PQ} = \frac{4-2}{3-(-1)} = \frac{2}{4} = \frac{1}{2}$ , $m_{QR} = \frac{0-4}{5-3} = \frac{-4}{2} = -2$ , $m_{RS} = \frac{-2-0}{1-5} = \frac{-2}{-4} = \frac{1}{2}$ , $m_{PS} = \frac{-2-2}{1-(-1)} = \frac{-4}{2} = -2$ . Since $m_{PQ} = m_{RS} = \frac{1}{2}$ , sides PQ and RS are parallel. Also, $m_{QR} = m_{PS} = -2$ , so QR and PS are parallel too. Choice B gives the correct slopes but for the wrong sides. Choices C and D have incorrect slope values.
Two lines have equations $3x + 4y = 12$ and $8x - 6y = 24$ . If a third line is perpendicular to the first line and parallel to a line with slope $\frac{4}{3}$ , what must be true about the relationship between all three lines?
Answer: The first and third lines are perpendicular, and the third line has the same slope as the second line First, convert the equations to slope-intercept form. For line 1: 3x + 4y = 12 → 4y = -3x + 12 → y = -3/4 x + 3, so slope₁ = -3/4. For line 2: 8x - 6y = 24 → -6y = -8x + 24 → y = 4/3 x - 4, so slope₂ = 4/3. The third line is perpendicular to the first line, so its slope must be the negative reciprocal of -3/4, which is 4/3. The third line is also parallel to a line with slope 4/3, which confirms its slope is 4/3. Since the second line also has slope 4/3, the second and third lines have the same slope. Therefore, the first and third lines are perpendicular, and the third line has the same slope as the second line.

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