![\"\"](\"https:\/\/gmatclub.com\/blog\/wp-content\/uploads\/2012\/05\/yx-300x300.png\" "\\"y=x\\"")
The 45\u00ba angle<\/h2>\nThe 45\u00ba angle<\/h2>\nFact<\/strong>: All lines with slopes of 1 make 45\u00ba angles with both the x- and y-axes.<\/p>\n Conversely, if a line makes a 45\u00ba angles with either the x- of y-axes, you know immediately its slope must be\u00a0 <\/p>\n Fact<\/strong>: Suppose we treat the line y = x as a mirror line.\u00a0\u00a0 If you take any point (a, b) in the coordinate plane, and reflect it over the line y = x, the result is (b, a).\u00a0 It reverses the x- and y-coordinates!<\/p>\n The corollary of this is that if we compare any two points with reversed coordinates, say (2, 7) and (7, 2), we automatically know that each is the image of the other by reflection over the line y = x.\u00a0 Add now the geometry fact that a mirror line is the set of all points equidistant from the original point and its image.\u00a0 This means that the midpoint of the segment connect (2, 7) and (7, 2) must lie on the line y = x.\u00a0 In fact, any point on the line y = x will be equidistant from both (2, 7) and (7, 2).\u00a0 Without doing a single calculation, we know, for example, that the triangle formed by, say, (2, 7) and (7, 2) and (8, 8 ) must be an isosceles triangle.<\/p>\n When we reflex over the line y = \u2013x, the coordinate are reversed and made their opposite sign: e.g. (2, 7) reflect to (\u20137, \u20132), and (\u20135, 3) reflects to (\u20133, 5).\u00a0 The other conclusions, about equidistance, remain the same.<\/p>\n <\/p>\n Fact<\/strong>: Any point (x, y) in the coordinate plane that is\u00a0above<\/strong>\u00a0the line y = x has the property that y > x.\u00a0 Any point (x, y) in the coordinate plane that is\u00a0below<\/strong>\u00a0the line y = x has the property that y < x.<\/p>\n Can you sense the veritable cornucopia of Data Sufficiency questions that could arise from this fact?\u00a0 If you every see a question about the coordinate plane asking whether y > x or y < x, chances are very good that the line y = x is hidden somewhere in the question.<\/p>\n <\/p>\n 1) Is the slope of Line 1 positive?<\/p>\n Statement #1: The angle between Line 1 and Line 2 is 40\u00ba. <\/p>\n 2) Point (P, Q) is in the coordinate plane.\u00a0 Is P > Q?<\/p>\n Statement #1: P is positive. <\/p>\n 3) A circle has a center at P = (\u20134, 4) and passes through the point (2, 3).\u00a0 Through which of the following must the point also pass?<\/p>\n 1) A straightforward prompt.<\/p>\n Statement #1 is intriguing: it gives us a specific angle measure.\u00a0 This is tantalizing, but unfortunately, it is only the angle between Line 1 and Line 2, and that angle could be oriented in any direction.\u00a0 Therefore, we can draw no conclusion about the prompt from this statement alone.\u00a0 Statement #1, by itself, is insufficient.<\/p>\n Statement #2 is also tantalizing, because it\u2019s numerically specific.\u00a0 But, unfortunately, this tells us a lot about Line 2 and zilch about Line one, so this statement is, by itself, is also insufficient.<\/p>\n Now, combine the statements.\u00a0 From statement #2, we know Line 2 has a slope of 1, which means the angle between Line 2 and the positive x-axis is 45\u00ba.\u00a0\u00a0 We know, from statement #1, that Line #1 is 40\u00ba away from Line 2.\u00a0 We don\u2019t know which way, above or below Line 2.\u00a0 If Line 1 is steeper than Line 2, it makes an angle of 45\u00ba + 40\u00ba = 85\u00ba with the positive x-axis.\u00a0 If Line 1 is less steep than Line 2, it makes an angle of 45\u00ba \u2013 40\u00ba = 5\u00ba with the positive x-axis.\u00a0 Either way, its angle above the positive x-axis is between 0\u00ba and 90\u00ba, which means it has a positive slope.\u00a0 The combined statements allow us to give a definitive answer to the prompt question.\u00a0 Answer =\u00a0C<\/strong>.<\/p>\n <\/p>\n 2) We see the x > y type question in the prompt, which makes us suspect that the line y = x will play an important part at some point.<\/p>\n Statement #1 just tells us P is positive, nothing else.\u00a0 The point (P, Q) = (4, 2) has the property that P > Q, but the point (P, Q) = (4, 5) has the property that P < Q.\u00a0 Clearly, just knowing P is positive does nothing to help us figure out whether P > Q.\u00a0 Statement #1, by itself, is wildly insufficient.<\/p>\n Statement #2 is intriguing.\u00a0 It discusses not the line y = x but the line y = x + 1.\u00a0 What is the relationship of those two lines?\u00a0 First of all, they are parallel: they have the same slope.\u00a0 The line y = x has a y-intercept of zero (it goes through the origin), while the line y = x + 1 has a y-intercept of 1.\u00a0 This means: any point on the line y = x + 1\u00a0must be above<\/em><\/strong>\u00a0the line y = x.\u00a0 If (P, Q) is on y = x + 1, then it is above y = x, which automatically means Q > P.\u00a0 We can give a definite \u201cno\u201d answer to the question.\u00a0 By itself, Statement #2 is sufficient.\u00a0 Answer =\u00a0B<\/strong>.<\/p>\n <\/p>\n 3) For this problem, there\u2019s a long tedious way to slog through the problem, and there\u2019s a slick elegant method that gets to the answer in a lightning fast manner.<\/p>\n The long slogging approach \u2014 first, calculate the distance from (\u20134, 4) to (2, 3).\u00a0 As it happens, that distance, the radius, equals\u00a0 The slick elegant approach is as follows.\u00a0 The point (\u20134, 4) is on the line y = \u2013x, so it is equidistant from any point and that point\u2019s reflection over the line y = \u2013x.\u00a0 The reflection of (2, 3) over the line y = \u2013x is (\u20133, \u20132).\u00a0 Since (\u20133, \u20132) is the same distance from (\u20134, 4) as is (2, 3), it must also be on the circle.\u00a0 Answer =\u00a0D<\/strong>.<\/p>\n This post was written by Mike McGarry, GMAT expert at\u00a0Magoosh<\/a>, and originally posted\u00a0here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" The 45\u00ba angle Fact: All lines with slopes of 1 make 45\u00ba angles with both the x- and y-axes. Conversely, if a line makes a 45\u00ba angles with either the…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,718,717,736],"tags":[],"class_list":["post-11756","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-data-sufficiency-gmat","category-problem-solving-gmat","category-quant-gmat","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11756","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/users\/133"}],"replies":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/comments?post=11756"}],"version-history":[{"count":2,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11756\/revisions"}],"predecessor-version":[{"id":11761,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/11756\/revisions\/11761"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=11756"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=11756"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=11756"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"pm](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_bed5862205e64706c5fa4d39046cbd02.png\" "\\"pm")
. This first fact is true, not only for y = x and y = \u2013x, for all lines of the form y = mx + b in which m equals either 1 or \u20131.\u00a0 If the slope is anything other than\u00a0, you would need trigonometry to figure out the angles, and that\u2019s beyond the scope of GMAT math.\u00a0 The GMAT could expect you to know this one fact about these special lines, especially on Data Sufficiency.<\/p>\nAs a Mirror<\/h2>\n
As a Boundary<\/h2>\n
Practice Questions<\/h2>\n
\nStatement #2: Line 2 has a slope of 1.<\/p>\n\n
\nStatement #2: Point (P, Q) above on the line y = x + 1.<\/p>\n\n
\n
<\/h2>\n
Practice Questions Explanations<\/h2>\n
![\"sqrt{37}\u00a0\"](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_6a53aafea69962af747c0939c081cc1b.png\" "\\"sqrt{37}\u00a0\\"")
.\u00a0 Then, we have to calculate the distance from\u00a0 (\u20134, 4) to each of the five answer choices, and find which one has also has a distance of \u00a0\u00a0\u2014- all without a calculator.\u00a0![\":(\"](\"https:\/\/magoosh.com\/gmat\/wp-includes\/images\/smilies\/icon_sad.gif\"){kind=icon}
<\/p>\n