71I'm taking it.. okay :) will post soon
6x + 8y = 48 intersects the coordinate axis at A and B respectively.
a line L bisects the area and the perimeter of the triangle AOB.
where O is the origin. the number of such lines possible is -
(a) 1
(b) 2
(c) 3
(d) more than 3.
slope of line L can be
(a) {10 + √56}/{10}
(b) {10 - √56}/{10}
(c) {8 + √36}/{10}
(d) none.
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UP 0 DOWN 0 0 3
3 Answers
3i read this on wikipedia.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other.
There are an infinitude of lines that bisect the area of a triangle. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are concurrent at the triangle's centroid; indeed, they are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides; each of these intersects the other two sides so as to divide them into segments with the proportions √2+1:1.
These six lines are concurrent three at a time: in addition to the three medians being concurrent, any one median is concurrent with two of the side-parallel area bisectors