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Home » Find the equations to the sides of the triangles the coordinates of whose angular points are respectively: (i) (1, 4), (2, -3) and (-1, -2) (ii) (0, 1), (2, 0) and (-1, -2)
Find the equations to the sides of the triangles the coordinates of whose angular points are respectively: (i) (1, 4), (2, -3) and (-1, -2) (ii) (0, 1), (2, 0) and (-1, -2)
CBSE | Class 11 | Exercise 23.5 | Maths | RD Sharma | Straight Lines
Find the equations to the sides of the triangles the coordinates of whose angular points are respectively: (i) (1, 4), (2, -3) and (-1, -2) (ii) (0, 1), (2, 0) and (-1, -2)

    \[~\left( \mathbf{i} \right)~\left( 1,\text{ }4 \right),\text{ }\left( 2,\text{ }-3 \right)\text{ }and\text{ }\left( -1,\text{ }-2 \right)\]

Given:

Points A (1, 4), B (2, -3) and C (-1, -2).

Let ,

    \[{{m}_{1,}}~{{m}_{2,}}~and\text{ }{{m}_{3}}~\]

be the slope of the sides AB, BC and CA,

respectively.

Hence,

The equation of the line passing through the two points:

    \[~\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }and~\left( {{x}_{2}},\text{ }{{y}_{2}} \right).\]

Then,

RD Sharma Solutions for Class 11 Maths Chapter 23 – The Straight Lines - image 12

    \[{{m}_{1\text{ }=}}~-7,~\]

    \[{{m}_{2}}~=\text{ }-1/3~\]

and

    \[{{m}_{3}}~=\text{ }3\]

therefore, the equation of the sides AB, BC and CA are

By using the formula,

    \[y\text{ }\text{ }{{y}_{1}}=\text{ }m\text{ }\left( x\text{ }\text{ }{{x}_{1}} \right)\]

    \[=>\text{ }y\text{ }\text{ }4\text{ }=\text{ }-7\text{ }\left( x\text{ }\text{ }1 \right)\]

or,

    \[y\text{ }\text{ }4\text{ }=\text{ }-7x\text{ }+\text{ }7\]

    \[y\text{ }+\text{ }7x\text{ }=\text{ }11,\]

    \[=>\text{ }y\text{ }+\text{ }3\text{ }=\text{ }\left( -1/3 \right)\text{ }\left( x\text{ }\text{ }2 \right)\]

or,

    \[3y\text{ }+\text{ }9\text{ }=\text{ }-x\text{ }+\text{ }2\]

or,

    \[3y\text{ }+\text{ }x\text{ }=\text{ }\text{ }7\]

    \[x\text{ }+\text{ }3y\text{ }+\text{ }7\text{ }=\text{ }0\]

and

    \[=>\text{ }y\text{ }+\text{ }2\text{ }=\text{ }3\left( x+1 \right)\]

or,

    \[y\text{ }+\text{ }2\text{ }=\text{ }3x\text{ }+\text{ }3\]

    \[y\text{ }\text{ }3x\text{ }=\text{ }1\]

now, we have

    \[y\text{ }+\text{ }7x\text{ }=11,\]

or,

    \[x+\text{ }3y\text{ }+\text{ }7\text{ }=0\]

    \[y\text{ }\text{ }3x\text{ }=\text{ }1\]

∴ The equation of sides are:

    \[y\text{ }+\text{ }7x\text{ }=11,\]

or,

    \[x+\text{ }3y\text{ }+\text{ }7\text{ }=0\]

or,

    \[y\text{ }\text{ }3x\text{ }=\text{ }1\]

 

 

    \[\left( \mathbf{ii} \right)~\left( 0,\text{ }1 \right),\text{ }\left( 2,\text{ }0 \right)\text{ }and\text{ }\left( -1,\text{ }-2 \right)\]

Given:

Points A (0, 1), B (2, 0) and C (-1, -2).

Let us assume,

    \[{{m}_{1,}}~{{m}_{2,}}~and\text{ }{{m}_{3}}\]

be the slope of the sides AB, BC and CA

hence,

The equation of the line passing through the two points :

    \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }and~\left( {{x}_{2}},\text{ }{{y}_{2}} \right).\]

Then,

RD Sharma Solutions for Class 11 Maths Chapter 23 – The Straight Lines - image 13

    \[{{m}_{1}}~=\text{ }-1/2,\]

next,

    \[{{m}_{2}}~=\text{ }-2/3~\]

and,

    \[{{m}_{3}}=\text{ }3\]

So, the equation of the sides AB, BC and CA are

According to formula,

    \[y\text{ }\text{ }{{y}_{1}}=\text{ }m\text{ }\left( x\text{ }\text{ }{{x}_{1}} \right)\]

or,

    \[=>\text{ }y\text{ }\text{ }1\text{ }=\text{ }\left( -1/2 \right)\text{ }\left( x\text{ }\text{ }0 \right)\]

or,

    \[2y\text{ }\text{ }2\text{ }=\text{ }-x\]

or,

    \[x\text{ }+\text{ }2y\text{ }=\text{ }2\]

or,

    \[=>\text{ }y\text{ }\text{ }0\text{ }=\text{ }\left( -2/3 \right)\text{ }\left( x\text{ }\text{ }2 \right)\]

or,

    \[3y\text{ }=\text{ }-2x\text{ }+\text{ }4\]

or,

    \[2x\text{ }\text{ }3y\text{ }=\text{ }4\]

or,

    \[=>\text{ }y\text{ }+\text{ }2\text{ }=\text{ }3\left( x+1 \right)\]

or,

    \[y\text{ }+\text{ }2\text{ }=\text{ }3x\text{ }+\text{ }3\]

or,

    \[y\text{ }\text{ }3x\text{ }=\text{ }1\]

now, we have

    \[x\text{ }+\text{ }2y\text{ }=\text{ }2,\]

or,

    \[2x\text{ }\text{ }3y\text{ }=4\]

or,

    \[y\text{ }\text{ }3x\text{ }=\text{ }1\]

∴ The equation of sides are:

    \[x\text{ }+\text{ }2y\text{ }=\text{ }2,\]

or,

    \[~2x\text{ }\text{ }3y\text{ }=4\]

    \[y\text{ }\text{ }3x\text{ }=\text{ }1\]

i

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