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Determine the vertices of the triangle formed by the lines $y = x, 3 y = x$ and $x + y = 8$ .

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(0, 0), (0, 4), (6, 2)

(0, 0), (4, 4), (0, 2)

(0, 0), (7, 7), (5, 2)

(0, 0), (4, 4), (6, 2)

answer is D.

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Detailed Solution

It is given the lines

y = x, 3 y = x

and

x + y = 8

.
The vertices of the triangle will be the point of intersection of given lines taken two at a time.
So, we need to find the points of intersection of the lines.
Therefore,

y = x

\dots (1)

3 y = x

\dots (2)

x + y = 8

\dots (3)

Taking (1) and (2),
Let, the point of intersection of above two lines is A.
Subtract

(1)

from

(2)

, we get,

2 y = 0

y = 0

Put this value of

y

in equation

(1)

, we get,

y = x = 0

Therefore, the coordinates of A are

(0, 0)

.
Taking (1) and (3)
Let, the point of intersection of above two lines is B.
Subtract

(1)

from

(3)

, we get,

x + y - y = 8 - x

2 x = 8

x = 4

Put this value of

x

in equation

(3)

, we get,

y = x = 4

Therefore, the coordinates of B are

(4, 4)

.
Taking (2) and (3)
Let, the point of intersection of above two lines is C,
Subtract equation

(2)

from

(3)

, we get,

x + y - x = 8 - 3 y

y = 8 - 3 y

4 y = 8

y = 2

Put this value of

y

in equation

(1)

, we get ,

x = 3 (2) = 6

So, the coordinates of C are

(6, 2)

.
Therefore, the coordinates of the triangle are

(0, 0), (4, 4)

and

(6, 2) .

Hence, correct option is 4.

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Determine the vertices of the triangle formed by the lines y=x,3y=x and x+y=8.

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Determine the vertices of the triangle formed by the lines $y = x, 3 y = x$ and $x + y = 8$ .