Question

If two vertices of an isosceles right triangle are

A(–1, –7) and B(–7, –1), then find coordinates of

third vertex of the triangle ABC [Given ∠C = 90°].​

Answer 1

Answer:

The coordinates of the third vertex of the isosceles right triangle will be ( - 7, - 7 ) OR ( - 1, - 1 ).

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given the coordinates of two vertices of an isosceles right triangle.

Let the isosceles right triangle be △ABC, ∠ C = 90°.

A ≡ ( - 1, - 7 ) ≡ ( x₁, y₁ )

B ≡ ( - 7, - 1 ) ≡ ( x₂, y₂ )

C ≡ ( x, y )

Now,

Slope of line AC = ( y - y₁ ) / ( x - x₁ )

⇒ Slope of line AC = [ y - ( - 7 ) ] / [ x - ( - 1 ) ]

⇒ Slope of line AC = ( y + 7 ) / ( x + 1 )

Now,

Slope of line BC = ( y - y₂ ) / ( x - x₂ )

⇒ Slope of line BC = [ y - ( - 1 ) ] / [ x - ( - 7 ) ]

⇒ Slope of line BC = ( y + 1 ) / ( x + 7 )

Now, we have given that,

AC ⊥ BC

We know that,

Product of slopes of two lines perpendicular to each other is - 1.

∴ Slope of line AC × Slope of line BC = - 1

⇒ ( y + 7 ) / ( x + 1 ) × ( y + 1 ) / ( x + 7 ) = - 1

⇒ ( y + 7 * y + 1 ) / ( x + 1 * x + 7 ) = - 1

⇒ ( y² + y + 7y + 7 ) / ( x² + 7x + x + 7 ) = - 1

⇒ ( y² + 8y + 7 ) / ( x² + 8x + 7 ) = - 1

⇒ y² + 8y + 7 = - 1 × ( x² + 8x + 7 )

⇒ y² + 8y + 7 = - x² - 8x - 7 - - ( 1 )

Now,

AC = BC - - [ Given ]

∴ √{ [ x - ( - 1 ) ]² + [ y - ( - 7 ) ]² } = √{ [ x - ( - 7 ) ]² + [ y - ( - 1 ) ]² }

⇒ √[ ( x + 1 )² + ( y + 7 )² ] = √[ ( x + 7 )² + ( y + 1 )² ]

⇒ ( x + 1 )² + ( y + 7 )² = ( x + 7 )² + ( y + 1 )² - - [ Squaring both sides ]

⇒ x² + 2x + 1 + y² + 14y + 49 = x² + 14x + 49 + y² + 2y + 1

⇒ 2x + 14y = 14x + 2y - - [ Cancelling the like terms ]

⇒ 14y - 2y = 14x - 2x

⇒ 12y = 12x

⇒ y = 12x / 12

⇒ y = x

∴ x = y - - ( 2 )

Now,

y² + 8y + 7 = - x² - 8x - 7 - - ( 1 )

⇒ x² + 8x + 7 = - x² - 8x - 7 - - [ From ( 2 ) ]

⇒ x² + 8x + 7 + x² + 8x + 7 = 0

⇒ 2x² + 16x + 14 = 0

⇒ 2x² + ( 14 + 2 ) x + 14 = 0

⇒ 2x² + 14x + 2x + 14 = 0

⇒ 2x ( x + 7 ) + 2 ( x + 7 ) = 0

⇒ ( x + 7 ) ( 2x + 2 ) = 0

⇒ x + 7 = 0 or 2x + 2 = 0

⇒ x = - 7 or 2x = - 2

⇒ x = - 7 or x = - 1

∴ x = y = - 7 or x = y = - 1

∴ The coordinates of the third vertex of the isosceles right triangle will be ( - 7, - 7 ) OR ( - 1, - 1 ).

Answer 2

$\huge\green{\mid{\fbox{\tt{Answer:-}}\mid}}$

(-1,-1) or (-7,-7)

$\huge\red{\mid{\fbox{\tt{Explanation:-}}\mid}}$

Here ABC is a right angle triangle given.

A(-1,-7) ,B(-7,-1),C(a,b) (suppose)

C=90° given it means AC=BC.draw a line that will be perpendicular to AB.Since ABC is isosceles triangle so its median and altitude is same.lets this is the point D that meets at AB.

This point D will be mid point of AB. Now get the coordinator of D using section formula.

D(-4,-4)

slope of AB

$= \frac{ - 7 + 1}{ - 1 + 7} = - 1$

So Solve Of CD = 1

Equation of line CD

$y + 4 = 1(x + 4)$

$x - y = 0$

It will passed from C, this line Satisfy

$a - b = 0$

$a = b.....(i)$

Now AC perpendicular on AB. So Multiple of Both Slope is equal to -1

$\frac{b + 1}{a + 7} \times \frac{b + 7}{a + 1} = - 1$

$(b + 1)(b + 7) + (a + 1)(a + 7 ) = 0$

$2(a + 1)(a + b) = 0$

a = -1 , a = -7

So cordinator of C can be

$\green\longrightarrow$(-1,-1) or (-7,-7)