Question

Aditya asked carpenter to make front of his guest house. The carpenter

suggested him a design which is plotted on a graph as shown in the below

figure:



(i) What is the length of the lines AB?

a) √10 units b)√11 units c) √13 units d) √14 units

(ii) The coordinates of the mid-point of BE are

a) (4,-3) b) (-3,4) c) (-3,-4) d) (-4,4)

(iii) Mid-point of ED will lie on:

a) x-axis b) y-axis c) x=y c)x=2y

(iv) If we join BD , then the y-axis divides BD in the ratio:

a) 1:1 b)1:2 c) 2:1 d)2:3 z​

Answer 1

Given :

A graph showing design of a guest house.

To find :

1. What is the length of line AB.
2. The coordinates of the mid point of BE are
3. Mid point of ED will lie on.
4. The ratio in which y axis divides the line segment BD.

Explanation :

If we consider any two points in a plane, $(x_{1} , y_{1} ) and (x_{2} , y_{2} )$ , then

• Length of the line is calculated using distance formula that is $\sqrt {(x_{2} - x_{1} )^{2} + (y_{2} - y_{1} )^{2}$ ; and
• Mid point of the line will be $\frac{x_{1} + x_{2} }{2} , \frac{y_{1}+y_{2} }{2}$ .
• Section formula = $\frac{m_{1} x_{2} + m_{2} x_{1} }{m_{1}+ m_{2} } , \frac{m_{1} y_{2}+m_{2}y_{1}}{m_{1} + m_{2} }$ where m₁:m₂ is the ratio in which the line is divided.

Solution 1:

From the figure, we can see the points A (0 ,8) and B (-3 ,6).

Applying distance formula, we get

$AB = \sqrt{(-3 - 0)^{2} + (6 -8)^{2} }$

      $= \sqrt{(9) + (4)}$

      $= \sqrt{13}$ units

Solution 2:

Let the mid point of B and E be P , then the coordinates of B and E are (-3 ,6) and (-3 ,2).

Using mid point formula, point P will be

$= \frac{(-3 + (-3))}{2} , \frac{ 6+ 2}{2}\\\\= (-3 , 4)$

Solution 3:

Let the mid point of E (-3 ,2) and D (3 ,2) be Q.

Hence, using mid point formula, Q will have coordinates

$= \frac{-3 +3}{2} , \frac{2 + 2}{2} \\= (0,2)$

Here, since x coordinate is 0, and y has a non zero value, the line joining ED will lie on y axis.

Solution 4:

We know the line BD cuts the y axis, hence at the point of intersection, the value of x coordinate will be 0.

Using this information.

Let the ratio be K:1 and we have the coordinates of B (-3 ,6) and D (3 ,2)

Therefore, using section formula.

$(0 , y) = \frac{k (3) + 1 (-3)}{k + 1} , \frac{k (2) + 1 (6)}{k + 1}$

For x coordinate,

$0 = \frac{3k - 3}{k + 1}$ ; or

3k - 3 = 0

k = 1

Hence, the ratio by which y axis divides BD is 1:1 .

Final answer :

Hence,

1. The length of the line AB is  option (C)$\sqrt{13}$ units.
2. The mid point of BE is option (B) (-3 ,4).
3. The mid point of ED will lie on y axis , option (B).
4. The y axis divides BD in the ratio 1:1 option (A).

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