Question

Find the nature of the triangle formed by the lines \[x + y - 4 = 0\], \[3x + y = 4\], \[x + 3y = 4\].
A. Isosceles
B. Equilateral
C. Right angled
D. None of these

Answer 1

Hint: First obtain the point of intersection of the given lines, then find the distance between the points \[(4,0),(1,1),(0,4)\]. Observe the distances and identify the triangle.

Formula used:
1. The distance between two points \[M(a,b)\] and \[N(c,d)\] is \[MN = \sqrt {{{\left( {a - c} \right)}^2} + {{\left( {b - d} \right)}^2}} \] .
2. If the lengths of three sides of a triangle are equal, the triangle is an equilateral triangle.
3. If the lengths of two sides of a triangle are equal, the triangle is an isosceles triangle.

Complete step by step solution:
The given lines are \[x + y - 4 = 0\], \[3x + y = 4\] and \[x + 3y = 4\].
Subtract \[x + y - 4 = 0\] from \[3x + y = 4\] we get,
\[3x + y - x - y = 4 - 4\]
\[2x = 0\]
\[x = 0\]
Put \[x = 0\] in \[3x + y = 4\] to obtain the first point.
\[3.0 + y = 4\]
\[y = 4\]
Hence the first point is \[A(0,4)\].

Multiply 3 by \[x + 3y = 4\] and subtract \[3x + y = 4\] from \[3x + 9y = 12\] we get,
\[3x + 9y - 3x - y = 12 - 4\]
\[8y = 8\]
\[y = 1\]
Put \[y = 1\] in \[3x + y = 4\] to obtain the second point.
\[3.x + 1 = 4\]
\[x = 1\]
Hence the second point is \[B(1,1)\].

The given lines are \[x + y - 4 = 0\], \[3x + y = 4\] and \[x + 3y = 4\].
Subtract \[x + y - 4 = 0\] from \[x + 3y = 4\] we get,
\[x + 3y - x - y = 4 - 4\]
\[2y = 0\]
\[y = 0\]
Put \[y = 0\] in \[x + y - 4 = 0\] to obtain the third point.
\[x + 0 - 4 = 0\]
\[x = 4\]
Hence the third point is \[C(4,0)\].
Distance between A and B is,
\[AB = \sqrt {{{\left( {4 - 1} \right)}^2} + {{(0 - 1)}^2}} \]
AB=\[\sqrt {{3^2} + 1} \]
\[AB = \sqrt {10} \]
Distance between B and C is,
\[BC = \sqrt {{{\left( {1 - 4} \right)}^2} + {{(1 - 0)}^2}} \]
BC =\[\sqrt {{3^2} + 1} \]
\[BC = \sqrt {10} \]
Distance between C and A is,
\[AC = \sqrt {{{\left( {4 - 0} \right)}^2} + {{(0 - 4)}^2}} \]
AC=\[\sqrt {{4^2} + {4^2}} \]
\[AC = \sqrt {32} \]
Therefore, the length of the two sides of the triangle is equal., so the triangle is isosceles.

The correct option is A.

Notes: Sometimes students get confused that how to obtain three vertices of the triangles, so the given lines are the equations of the sides of the triangle, therefore the intersecting points of any two lines give a vertex of the triangle.