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Two farmers, X and Y, each surround a triangular plot of land with a fence on their respective properties. Farmer X requires 240 meters of fencing and farmer Y requires 60 meters of fencing. If the ratios of the lengths of the corresponding sides of the triangular plots of land are all equal to k, then the area of the triangular plot of land on farmer X‘s property is how many times bigger than the area of the triangular plot of land on farmer Y‘s property?
A. 2
B. 2k
C. 8
D. 8k
E. 16
Two farmers, X and Y, each surround a triangular plot of land with a fence on their respective properties. Farmer X requires 240 meters of fencing and farmer Y requires 60 meters of fencing. If the ratios of the lengths of the corresponding sides of the triangular plots of land are all equal to k, then the area of the triangular plot of land on farmer X‘s property is how many times bigger than the area of the triangular plot of land on farmer Y‘s property?
A. 2
B. 2k
C. 8
D. 8k
E. 16
Taking ratios as
\(\frac{A}{a} = \frac{B}{b} = \frac{C}{c} = k\) where A,,B and C are sides of triangular plot of farmer X and a,b and c are sides of triangular plot of farmer Y.
A = ak, B = bk and C = ck
Also, a + b + c = 60 and A + B + C = 240
Now, A + B + C = 240
ak + bk + ck = 240
(a + b + c)k = 240
k = 4
Thus sides of farmer X's plot is 4 times the corresponding sides of farmer Y's plot.
From here we may assume triangular plots be of any shape with one being scaled up version of the other.
Let's take a right angled triangle with perpendicular equal sides as a and b or A and B with c and C as hypotenuse(though hardly matters) respectively.
Now, Area of triangular plot of farmer X = \(\frac{1}{2} * A * B\)
and
Area of triangular plot of farmer Y = \(\frac{1}{2} * a * b\)
So, \(\frac{Area of triangular plot of farmer X}{Area of triangular plot of farmer Y} = \frac{1}{2} * A * B / \frac{1}{2} * a * b\)
= \(\frac{4a * 4b }{ a * b}\)
= 16
Answer E.
A. 2
B. 2k
C. 8
D. 8k
E. 16
Taking ratios as
\(\frac{A}{a} = \frac{B}{b} = \frac{C}{c} = k\) where A,,B and C are sides of triangular plot of farmer X and a,b and c are sides of triangular plot of farmer Y.
A = ak, B = bk and C = ck
Also, a + b + c = 60 and A + B + C = 240
Now, A + B + C = 240
ak + bk + ck = 240
(a + b + c)k = 240
k = 4
Thus sides of farmer X's plot is 4 times the corresponding sides of farmer Y's plot.
From here we may assume triangular plots be of any shape with one being scaled up version of the other.
Let's take a right angled triangle with perpendicular equal sides as a and b or A and B with c and C as hypotenuse(though hardly matters) respectively.
Now, Area of triangular plot of farmer X = \(\frac{1}{2} * A * B\)
and
Area of triangular plot of farmer Y = \(\frac{1}{2} * a * b\)
So, \(\frac{Area of triangular plot of farmer X}{Area of triangular plot of farmer Y} = \frac{1}{2} * A * B / \frac{1}{2} * a * b\)
= \(\frac{4a * 4b }{ a * b}\)
= 16
Answer E.
