Name: Angle Between Two Lines
Uploaded: 2019-07-19
Duration: 6 min 13 s
Description: If two lines are perpendicular to each other, their direction vectors are also perpendicular. This means the scalar product of the direction vectors is equal...

Question

Two sides of an isosceles triangle are given by the equations

7 x - y + 3 = 0

and

x + y - 3 = 0

and its third side passes through the point

(1, - 10)

. Determine the equation of the third side.

Accepted Answer

Let $A B C$ be the isosceles triangle, where $7 x - y + 3 = 0$ and $x + y - 3 = 0$ represent the sides $A B$ and $A C$ , respectively.
Let $A B = B C$

$∵ A B = B C$

$\Rightarrow ∠ B = ∠ C$ [Since angles opposite to equal sides are equal].

$∴ \tan B = \tan C$

Here,

Slope of $A B = 7$

Slope of $A C = - 1$

Let $m$ be the slope of $B C$ .

Then, $|\frac{m - 7}{1 + 7 m}| = |\frac{m + 1}{1 - m}| = |\frac{m + 1}{m - 1}|$

$\Rightarrow \frac{m - 7}{1 + 7 m} = \pm \frac{m + 1}{m - 1}$

Taking the positive sign, we get:

$m^{2} - 8 m + 7 = 7 m^{2} + 8 m + 1$

$\Rightarrow (m + 3) (m - \frac{1}{3}) = 0$

$\Rightarrow m = - 3, \frac{1}{3}$

Now, taking the negative sign, we get:

$(m - 7) (m - 1) = - (7 m + 1) (m + 1)$

$\Rightarrow m^{2} - 8 m + 7 = - 7 m^{2} - 8 m - 1$

$\Rightarrow m^{2} = - 1$ (not possible)

Equations of the third side is,

$y + 10 = - 3 (x - 1)$ and $y + 10 = \frac{1}{3} (x - 1)$

$\Rightarrow 3 x + y + 7 = 0$ and $x - 3 y - 31 = 0$

Two sides of an isosceles triangle are given by the equations $7 x - y + 3 = 0$ and $x + y - 3 = 0$ and its third side passes through the point $(1, - 10)$ . Determine the equation of the third side.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Two sides of an isosceles triangle are given by the equations 7x-y+3=0 and x+y-3=0 and its third side passes through the point (1,-10). Determine the equation of the third side.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Two sides of an isosceles triangle are given by the equations $7 x - y + 3 = 0$ and $x + y - 3 = 0$ and its third side passes through the point $(1, - 10)$ . Determine the equation of the third side.