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Bunuel
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The line 12x + 5y = 60 forms a triangle with the coordinate axes. What 25 Apr 2019, 03:09
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The line 12x + 5y = 60 forms a triangle with the coordinate axes. What is the sum of the lengths of the three altitudes of this triangle?

(A) 20
(B) 360/17
(C) 107/5
(D) 43/2
(E) 281/13
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The line 12x + 5y = 60 forms a triangle with the coordinate axes. What 26 Apr 2019, 06:59
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here the line will intersect x axis at (5,0) and y axis at (12,0) ( put x = 0 to find the cut point at x axis and put y=0 to find the cut point at y axis )

So the two altitude will be 12 and 5
and the third will be from center (0,0) to the given line = (|12x+5y-60|)/\sqrt{(12^2+5^2)}

put x = 0 and y = 0
we get 60/13

so the sm of altitudes = 5+12+60/13 = 281/13


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The line 12x + 5y = 60 forms a triangle with the coordinate axes. What 19 Sep 2020, 00:14
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m1033512, How did you get the value of the altitude?
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Akshat1994
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The line 12x + 5y = 60 forms a triangle with the coordinate axes. What 21 Sep 2020, 00:45
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Perpendicular distance from a point (m,n) to a line ax+by+c=0 is given by:

|(am+bn+c)| / (sq rt. of (a^2+ b^2)

Substitute (0,0) in |12x+5y-60|= |-60|= 60 (positive since it is distance)

Perpendicular distance = 60/ sq rt (12^2 + 5^2)= 60/13

Other two altitudes = 12 and 5
Sum of altitudes = 12+5+ 60/13 = 281/13

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The line 12x + 5y = 60 forms a triangle with the coordinate axes. What 24 Sep 2020, 00:09
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Trick and Concept: Most of the time (on the GMAT at least), when a Right Triangle is Involved, the GMAT will be testing 1 of 4 triangles: 30/60/90 ; 45/45/90 ; 3-4-5 ; 5-12-13


Another Concept: Given the 2 Legs of a Right Triangle and you need to find the Altitude from the 90 Degree Vertex to the Hypotenuse as the Base, find the Area first, then set the Area you find equal to = 1/2 * Hypotenuse * Altitude from 90 Degree Vertex


1st) The Triangle Crosses the Y-Axis at (0 , 12) ---- found by finding the Value of Y when X = 0 in the Equation

Similarly, the triangle crosses the X-Axis at (5 , 0) ---- found by finding the Value of X when Y = 0 in the Equation


2nd) We are told that the Line forms a Triangle with the 2 Axes. Using the Origin as a Vertex, the Line creates a Triangle with Legs of Length 5 Units and 12 Units. Therefore the Hypotenuse from Point (0 , 12) to Point (5 , 0) must = 13 Units


3rd) Area of Triangle = 1/2 * Leg * Leg = 1/2 * 12 * 5 = 30


Using the Hypotenuse = 13 as the BASE

Let the Altitude from the 90 Degree Vertex at the Origin = H

Area of Triangle = 30 = 1/2 * (13) * H

H = Altitude from the 90 Degree Vertex = 60/13


4th) the Other 2 Altitudes will be the 2 Legs of a Right Triangle ---- 5 and 12


SUM of all 3 Altitudes = 60/13 + 5 + 12 = 221/13 + 60/13 = 281/13

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