Question

11. Solve the following:

1. Find the distance between the pair of points (1.2) and (4,3)
2. Determine whether the given set of points are collinear or not
A (7-2), B(5,1),C (34).
3. Show that the following points form an isosceles triangle
A(5,4), B(2.0), C(-2,3).
4. Show that the following points from the vertices of a parallelogram
A(-3,1), B(-6-7). CO-9.046,-1).
5. The abscissa of a point A is equal to its ordinate, and its distance from the point (1.3) is 10
units, what are the coordinates of A?​

Answer 1

Step-by-step explanation:

(i) Here x1 = 2, y1 = 3, x2 = 4 and y2 = 1

∴ The required distance

Answer 2

Step-by-step explanation:

1) The distance between two points (1,2) and (4,3) is given by : distance

$d = \sqrt{ {(x2 - x1)}^{2} + {(y2 - y1}^{2} )}$

Here (x1 , y1 ) = (1,2) and ( x2 , y2 ) = (4,3)

$d = \sqrt{ ({4 - 1}^{2}) + ( {3 - 2}^{2} ) }$

$d = \sqrt{ {3}^{2} + {1}^{2} }$

$d = \sqrt{9 + 1}$

$d = \sqrt{10}$

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ii) Here ( x1 , y1 ) = (7-2) , ( x2 , y2 ) = ( 5, 1 ) and ( x3 , y3 ) = ( 3 , 4 )

Therefore using :

$\frac{1}{2} \sqrt{x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)}$

$\frac{1}{2} \sqrt{7(1 - 4)) + 5(4 -( - 2)) + 3( - 2 - 1) }$

$\frac{1}{2} \sqrt{7 \times -3 + 5 \times 6 + 3 \times - 3}$

$\frac{1}{2} \sqrt{ - 21 + 30 - 9}$

$\frac{1}{2} \sqrt{9 - 9}$

$\frac{1}{2} \sqrt{0}$

$= 0$

This is equal to 0 so thus points are collinear.

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iii) To show the points A, B and C will make an isosceles triangle, we need to show the length of any two of them equal.

Now ,

AB :

$= \sqrt{ ({5 - 2)}^{2} + ( {4 - 0)}^{2} }$

$= \sqrt{9 + 16}$

$= \sqrt{25}$

$= 5 \: units$

BC :

$= \sqrt{( {2 + 2)}^{2} + ( {0 - 3}^{2}) }$

$= \sqrt{16 + 9}$

$= \sqrt{25}$

$= 5 \: units$

CA :

$= \sqrt{( { - 2 - 5}^{2}) + ( {3 - 4)}^{2} }$

$= \sqrt{49 + 1}$

$= \sqrt{50}$

$= 5 \sqrt{2 \: } units$

Since AB = BC so , triangle ABC is an isosceles triangle.

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4.) sorry but values are not properly written.. u can refer to the above attachement and solve.^_^

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5.) Answer is in the above attachement ..

I hope it helps u dear friend ^_^!!!!

Have a nice day ♡♡