Question

Find the length of median BD of ABC, if A(7.-3), B(5.3) and C(3.1).​

Answer 1

Answer:

The length of diagonal BD is 2 units. ................

Answer 2

Answer:

The length of the median BD is 4 units.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given the coordinates of a triangle.

We have to find the length of a median of the triangle.

$\bullet\sf\:A\:\equiv\:(\:x_1\:,\:y_1\:)\:\equiv\:(\:7\:,\:-\:3\:)\\\\\\\bullet\sf\:B\:\equiv\:(\:x_2\:,\:y_2\:)\:\equiv\:(\:5\:,\:3\:)\\\\\\\bullet\sf\:C\:\equiv\:(\:x_3\:,\:y_3\:)\:\equiv\:(\:3\:,\:1\:)\\\\\\\bullet\sf\:D\:\equiv\:(\:x\:,\:y\:)$

In figure, segment BD is a median.

Hence, D is the midpoint of AC.

We know that,

$\pink{\sf\:x\:=\:\dfrac{x_1\:+\:x_2}{2}}\sf\:\:\:,\:\:\:\pink{y\:=\:\dfrac{y_1\:+\:y_2}{2}}\sf\:\:\:-\:-\:[\:Formula\:]\\\\\\\implies\sf\:x\:=\:\dfrac{x_1\:+\:x_3}{2}\:\:\:,\:\:\:y\:=\:\dfrac{y_1\:+\:y_3}{2}\:\:\:-\:-\:[\:Coordinates\:of\:C\:]\\\\\\\implies\sf\:x\:=\:\dfrac{7\:+\:3}{2}\:\:\:,\:\:\:y\:=\:\dfrac{-\:3\:+\:1}{2}\\\\\\\implies\sf\:x\:=\:\cancel{\frac{10}{2}}\:\:\:,\:\:\:y\:=\:-\:\cancel{\frac{2}{2}}\\\\\\\implies\boxed{\red{\sf\:x\:=\:5}}\sf\:\:\:,\:\:\:\boxed{\red{\sf\:y\:=\:-\:1}}$

Now, we have to find the length of segment BD.

$\pink{\sf\:d\:(\:B\:,\:D\:)\:=\:\sqrt{\:(\:x_2\:-\:x\:)^2\:+\:(\:y_2\:-\:y\:)^2\:}}\sf\:\:\:-\:-\:[\:Formula\:]\\\\\\\implies\sf\:d\:(\:B\:,\:D\:)\:=\:\sqrt{\:(\:5\:-\:5\:)^2\:+\:[\:3\:-\:(\:-\:1\:)\:]^2\:}\\\\\\\implies\sf\:d\:(\:B\:,\:D\:)\:=\:\sqrt{\:(\:0\:)^2\:+\:(\:3\:+\:1\:)^2\:}\\\\\\\implies\sf\:d\:(\:B\:,\:D\:)\:=\:\sqrt{\:0\:+\:(\:4\:)^2\:}\\\\\\\implies\sf\:d\:(\:B\:,\:D\:)\:=\:\sqrt{\:16\:}\\\\\\\implies\boxed{\red{\sf\:d\:(\:B\:,\:D\:)\:=\:4\:units}}$

Additional Information:

1. Distance Formula:

The formula which is used to find the distance between two points using their coordinates is called distance formula.

$\large{\boxed{\red{\sf\:d\:(\:A\:,\:B\:)\:=\:\sqrt{\:(\:x_{1}\:-\:x_{2}\:)^{2}\:+\:(\:y_{1}\:-\:y_{2}\:)^{2}\:}}}\:\:}$

2. Section Formula:

The formula which is used to find the coordinates of a point which divides a line segment in a particular ratio is called section formula.

$\large{\boxed{\red{\sf\:x\:=\:\dfrac{mx_{2}\:+\:nx_{1}\:}{m\:+\:n}}}}\:\:\sf\:\&\:\:\:\large{\boxed{\red{\sf\:y\:=\:\dfrac{my_{2}\:+\:ny_{1}\:}{m\:+\:n}}}}$

3. Midpoint formula:

If the point divides a line segment into two equal parts, then the formula used to find the coordinates of that point is called midpoint formula.

$\large{\boxed{\red{\sf\:x\:=\:\dfrac{x_1\:+\:y_1}{2}}}}\sf\:\:\:\&\:\:\:\boxed{\red{\sf\:y\:=\:\dfrac{y_1\:+\:y_2}{2}}}$