Question

1.the distance between the points (1,2) and (x, - 1) is 5 unit. find a possible value of x
2. Show that p(2,3), Q (- 3, - 2) and r(2, -7) are the vertical of an isosceles triangle

Answer 1

Answer:

Value of x is 5 and - 3.

Given points are forming a isosceles triangle.

Step-by-step explanation:

Question 1

From the identities of coordinate geometry.

Distance between two points = √{( a - x )^2 + ( b - y )^2 }, where we have to find the distance between points ( a , b ) and ( x , y ).

The above given formula is also known as distance formula.

Here,

Given points are : ( 1 , 2 ) and ( x , - 1 ), and distance between them is of 5 units.

= > Distance between the given points = 5units

= > √[ ( 1 - x )^2 + { 2 - ( - 1 ) }^2 ] = 5

= > ( 1 - x )^2 + ( 2 + 1 )^2 = 5^2

= > ( 1 - x )^2 + ( 3 )^2 = 5^2

= > ( 1 - x )^2 = 25 - 9

= > ( 1 - x )^2 = 16 = ( ±4 )^2

= > 1 - x = ± 4

If 1 - x is equal to - 4 :

= > 1 - x = - 4

= > 1 - ( - 4 ) = x

= > 1 + 4 = x

= > 5 = x

If 1 - x is equal to 4 :

= > 1 - x = 4

= > 1 - 4 = x

= > - 3 = x

Hence the possible value(s) of x are 5 and - 3.

Question 2

If the given points are forming an isosceles triangles, then the length of two sides should be equal.

Thus,

= > Length of side pQ = Distance between p and Q

= > Length of side pQ = √[ ( - 3 - 2 )^2 + ( - 2 - 3 )^2 ]

= > Length of side pQ = √[ ( - 5 )^2 + ( - 5 )^2 ]

= > Length of side pQ = √( 25 + 25 ) = 5√2 units

Thus, length of side pQ is 5√2 units.

= > Length of side Qr = Distance between r and Q

= > Length of side Qr = √[ ( - 2 - 2 )^2 + { - 2 - ( - 7 )}^2 ]

= > Length of side Qr = √[ ( - 5 )^2 + ( - 2 + 7 ) }^2 ]

= > Length of side Qr = √[ ( - 5 )^2 + ( 5 )^2 ]

= > Length of side Qr = √( 25 + 25 ) = 5√2 units

Thus, length of side Qr is 5√2 units.

= > Length of side rp = Distance between r and p

= > Length of side rp = √[ ( - 3 - 2 )^2 + ( - 7 - 3 )^2 ]

= > Length of side rq = √[ ( 0 )^2 + ( - 10 )^2 ]

= > Length of side rq = √( 100 )

Thus, length of side rq is 10.

Since two sides are equal, given points are forming an isosceles triangle.