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Parallel lines and also their slopes room easy. Because slope is a measure of the edge of a heat from the horizontal, and since parallel currently must have the same angle, climate parallel lines have the same slope — and also lines with the same slope room parallel.

You are watching: If two lines are perpendicular their slopes are negative reciprocals

Perpendicular lines space a bit an ext complicated.

If friend visualize a line with optimistic slope (so it"s an increasing line), then the perpendicular line must have an unfavorable slope (because that will have to be a to decrease line). For this reason perpendicular lines have actually slopes which have opposite signs.


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

To provide a numerical instance of "negative reciprocals", if the one line"s steep is m = 4/5, then the perpendicular line"s slope will certainly be m = –5/4. Or, if the one line"s steep is m = –2, then the perpendicular line"s slope will be m = 1/2. (Remember that any integer have the right to be turned into a fraction by putting it over 1.)


In your homework, girlfriend will more than likely be given some bag of points, and be asked come state even if it is the lines v the pairs 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 (1,2); one more line passes with the points (–2,0) and also (0,4). Space these currently parallel, perpendicular, or neither?

To price this question, I"ll find the 2 slopes. For this reason I have the right to keep things straight and tell the difference between the two slopes, I"ll use subscripts.


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One heat passes v the points (0,–4) and also (–1,–7); an additional line passes through the points (3,0) and (–3,2). Room these lines parallel, perpendicular, or neither?

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If ns were to transform the "3" come fractional kind by placing it end "1", climate flip the and readjust its sign, ns would obtain "–1/3" . This negative reciprocal of the very first slope matches the value of the second slope. In various other words, these slopes are negative reciprocals, so:


One heat passes through the points (–4,2) and also (0,3); an additional line passes with the points (–3,–2) and (3,2). Are these lines parallel, perpendicular, or neither?

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These slope values are not the same, for this reason the lines space not parallel. The slope values are likewise not an unfavorable reciprocals, therefore the lines room not perpendicular. Climate the answer is:


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

This slope have the right to be turned into a fraction by placing it end 1, so this slope have the right to be restated as:


To gain the an unfavorable reciprocal, I must flip this fraction, and change the sign. Then the steep of any type of line perpendicular come the provided line is:


Warning: as soon as asked a concern of this form ("are these lines parallel or perpendicular?"), perform not start drawing pictures. If the lines space close to being parallel or close to being perpendicular (or if you draw the lines messily), you deserve to very-easily obtain the dorn answer from her picture.

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