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17 Sep 2021, 01:15
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How many triangles with perimeter 15 can be formed such that all three sides are integer?
A. 7
B. 6
C. 8
D. 5
E. 4
A. 7
B. 6
C. 8
D. 5
E. 4
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17 Sep 2021, 02:19
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Bunuel
How many triangles with perimeter 15 can be formed such that all three sides are integer?
A. 7
B. 6
C. 8
D. 5
E. 4
A. 7
B. 6
C. 8
D. 5
E. 4
Number of triangle when perimeter is given.
if perimeter is even ,Number of triangles =\( \frac{(p^2)}{48}\)
If perimeter is odd,Number of triangles =\( \frac{((p+3)^2)}{48}\)
Here we will apply second one, and number of triangles =\( \frac{18^2}{48}\) = 6.xx =6
IMO B
07 May 2022, 03:53
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Let us assume that the three sides of the triangle be x,y and z.
We have the perimeter = 15
Therefore, x + y + z = 15
For the triangle to hold,
z < x + y
y< x + z
x< y+ z
Let us take any one side,
z = 15 - (x + y)
Also,
z < x+ y
15 - (x + y) < x + y
15 < 2(x+y)
Therefore,
x+y > 7.5
Since x and y are integers, the only values (x + y) can take are 8,9,10,11,12,13 and 14 (Since z is a positive integer)
Hence 7 triangles can be formed
We have the perimeter = 15
Therefore, x + y + z = 15
For the triangle to hold,
z < x + y
y< x + z
x< y+ z
Let us take any one side,
z = 15 - (x + y)
Also,
z < x+ y
15 - (x + y) < x + y
15 < 2(x+y)
Therefore,
x+y > 7.5
Since x and y are integers, the only values (x + y) can take are 8,9,10,11,12,13 and 14 (Since z is a positive integer)
Hence 7 triangles can be formed
