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Parallel lines and also their slopes are easy. Due to the fact that slope is a measure up of the edge of a line from the horizontal, and since parallel lines must have actually the very same angle, climate parallel lines have actually the very same slope — and lines through the exact same slope room parallel.

You are watching: Two lines are parallel if their slopes are

Perpendicular lines space a bit more complicated.

If you visualize a heat with optimistic slope (so it"s an enhancing line), then the perpendicular line must have negative slope (because that will need to be a to decrease line). Therefore perpendicular lines have slopes which have actually opposite signs.


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The other "opposite" thing with perpendicular slopes is that their values are reciprocals; the is, you take it the one steep value, and flip the upside down. (This is the non-obvious thing around the slopes the perpendicular lines.) placed this together with the authorize change, and you get that the slope of a perpendicular heat is the "negative reciprocal" the the steep of the initial line — and also two lines through slopes that are negative reciprocals that each other are perpendicular to every other.

To give a numerical instance of "negative reciprocals", if the one line"s slope is m = 4/5, climate the perpendicular line"s slope will be m = –5/4. Or, if the one line"s steep is m = –2, climate the perpendicular line"s slope will be m = 1/2. (Remember that any integer can be turned right into a portion by placing it end 1.)


In her homework, you will probably be given some pairs of points, and also be asked to state whether the lines through the bag of points room "parallel, perpendicular, or neither". To answer the question, you"ll need to calculate the slopes and also compare them. Here"s just how that works:

One heat passes with the points (–1,–2) and also (1,2); another line passes v the points (–2,0) and (0,4). Room these present parallel, perpendicular, or neither?

To answer this question, I"ll uncover the two slopes. For this reason I can keep points straight and also tell the difference between the 2 slopes, I"ll usage subscripts.


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One line passes v the clues (0,–4) and (–1,–7); an additional line passes with the points (3,0) and (–3,2). Are these present parallel, perpendicular, or neither?

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If i were to transform the "3" come fractional form by putting it end "1", climate flip the and readjust its sign, i would acquire "–1/3" . This an adverse reciprocal the the first slope matches the value of the second slope. In various other words, this slopes are an adverse reciprocals, so:


One heat passes with the clues (–4,2) and (0,3); another line passes v the point out (–3,–2) and also (3,2). Room these present parallel, perpendicular, or neither?

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These slope values space not the same, therefore the lines are not parallel. The slope values are likewise not an adverse reciprocals, therefore the lines are not perpendicular. Climate the prize is:


They"ve offered me the original line"s equation, and it"s in "y=" form, so it"s easy to discover the slope. I deserve to just check out the value off the equation: m = –4.

This slope deserve to be turned into a fraction by putting it end 1, for this reason this slope can be restated as:


To gain the negative reciprocal, I must flip this fraction, and adjust the sign. Climate the steep of any line perpendicular come the offered line is:


Warning: once asked a question of this type ("are these lines parallel or perpendicular?"), perform not start drawing pictures. If the lines room close to being parallel or near to gift perpendicular (or if you attract the present messily), you deserve to very-easily acquire the dorn answer from your picture.

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Besides, they"re no asking if the present look parallel or perpendicular; they"re asking if the lines in reality are parallel or perpendicular. The only means to be certain of your answer is to perform the algebra.