Question

Find the length of median of triangle ABC drawn from A, where A(4,-8), B(3,6) and C(5,-4) are vertices of triangle . ​

Answer 1

$\large\text{\underline{Required}}$

The length of the median drawn from A.

$\large\text{\underline{Given}}$

The coordinates of the three points.

• $A(4,-8)$
• $B(3,6)$
• $C(5,4)$

$\large\text{\underline{Solution}}$

The median bisects the opposite segment from $\angle A$. This means we need to find the median to $\overline{BC}$.

Let the midpoint be $M$. Then

$\large\underline{\text{Step 1. Midpoint formula}}$

If the segment joining $P_{1}(x_{1},y_{1})$ and $P_{2}(x_{2},y_{2})$ is bisected, the midpoint lies on the coordinate $(\dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2})$.

$\red{\bigstar}(\dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2})\text{ is the midpoint of segment joining two points.}$

Now, according to the midpoint formula,

$\implies M(\dfrac{3+5}{2},\dfrac{6-4}{2})$

$\implies M(4,1)$

$\large\underline{\text{Step 2. Midpoint formula}}$

The length of the segment joining $P_{1}(x_{1},y_{1})$ and $P_{2}(x_{2},y_{2})$ is equal to the following.

$\red{\bigstar}\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}\text{ is the distance between two points.}$

The length of the median is $\overline{AM}$. Now we can use distance formula between two points.

$\implies\overline{AM}=\sqrt{(4-4)^{2}+(-8-1)^{2}}$

$\implies\overline{AM}=\sqrt{9^{2}}$

$\implies\overline{AM}=9$

$\large\text{\underline{Learn more}}$

Picture included in the attachment.

$\red{\bigstar}\overline{AB}^{2}+\overline{AC}^{2}=2(\overline{AM}^{2}+\overline{BM}^{2})\text{ (Apollonius's Theorem)}$

This formula is for the length of the median.

We can also verify our answer with this formula.

$\implies\overline{AB}^{2}=(4-3)^{2}+(-8-6)^{2}=197$

$\implies\overline{AC}^{2}=(4-5)^{2}+(-8+4)^{2}=17$

$\implies\overline{BM}^{2}=(3-4)^{2}+(6-1)^{2}=26$

This gives the following answer.

$\implies\overline{AM}^{2}=81$

$\implies\overline{AM}=9$

Answer 2

Step-by-step explanation:

solution :

• AD= 81

• BE= 9

• hope it helps you otherwise you have any problem in this question please ask me