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In the xy-plane, the coordinates of the three vertices of a triangle a
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18 Oct 2021, 05:17
18 Oct 2021, 05:17
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In the xy-plane, the coordinates of the three vertices of a triangle are given by (0, 0), (0, a), and (b, 0). The triangle’s area is 6, and all three of its sides have integer length. What is the triangle’s perimeter?
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Re: In the xy-plane, the coordinates of the three vertices of a triangle a
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19 Oct 2021, 13:11
19 Oct 2021, 13:11
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Carcass wrote:
In the xy-plane, the coordinates of the three vertices of a triangle are given by (0, 0), (0, a), and (b, 0). The triangle’s area is 6, and all three of its sides have integer length. What is the triangle’s perimeter?
Based on the three coordinates (0,0), (0,a) and (b,), the triangle is right-angled and area is \(\frac{(ab)}{2} = 6\)
and \(ab = 12\)
to meet the criteria "all three of its sides have integer length", the only option for \(a = 3 or 4\) and \(b = 4 or 3\)
In either case, the perimeter is same = 12
In the xy-plane, the coordinates of the three vertices of a triangle a
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06 May 2023, 22:33
06 May 2023, 22:33
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Beauty of this problem is all points are not in Same plane
The formula to calculate the distance between two points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) is
d=(x1−x2)2+(y1−y2)2−−−−−−−−−−−−−−−−−−−√d=(x1−x2)2+(y1−y2)2.
Step 1) Distance between (0,0) and (0,a) using above formula = a
Step 2) Distance between (0,0) and (b,0) using above formula = b ( it could be height * because P(0,0) & (0,b) on same plane,x co ordinate is 0 fr both )
Drop a vertical line from (0,b) to bottom line to get height
Area = ab/2 = 6 => ab = 12 => a or b could be 3 , 4 or 4,3
c = Squart of ( 9 + 16) = 5
So, Perimeter = a+ b+ c side = 3+4+5 = 12
The formula to calculate the distance between two points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) is
d=(x1−x2)2+(y1−y2)2−−−−−−−−−−−−−−−−−−−√d=(x1−x2)2+(y1−y2)2.
Step 1) Distance between (0,0) and (0,a) using above formula = a
Step 2) Distance between (0,0) and (b,0) using above formula = b ( it could be height * because P(0,0) & (0,b) on same plane,x co ordinate is 0 fr both )
Drop a vertical line from (0,b) to bottom line to get height
Area = ab/2 = 6 => ab = 12 => a or b could be 3 , 4 or 4,3
c = Squart of ( 9 + 16) = 5
So, Perimeter = a+ b+ c side = 3+4+5 = 12
