Chapter 9.5, Problem 6PSA

a.

To determine

To calculate: The length of a median from a point.

Answer to Problem 6PSA

The length of a median from a point is $\sqrt{85}$ .

Explanation of Solution

Given information:

R = $\left(1,-4\right),T=\left(9,8\right)$ and V = $\left(13,2\right)$

Formula used:

The distance between two points by using the distance formula, which is an application of the Pythagoreantheorem.

D $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Calculation:

  

R = $\left(1,-4\right),T=\left(9,8\right)$ and V = $\left(13,2\right)$

Midpoint $\overline{RV}=(7,-1)$

Distance from T to midpoint $\overline{RV}$

  $\begin{array}{l}=\sqrt{{\left(9-7\right)}^2+{\left(8-(-1)\right)}^2}\\ =\sqrt{4+81}\\ =\sqrt{85}\end{array}$

b.

To determine

To find: The length of the segment.

Answer to Problem 6PSA

The length of the segment is $3\sqrt{5}$ .

Explanation of Solution

Given information:

R = $\left(1,-4\right),T=\left(9,8\right)$ and V = $\left(13,2\right)$

Formula used:

The distance between two points by using the distance formula, which is an application of the Pythagoreantheorem.

D $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Calculation:

  

The length of the segment joining the midpoints of two sides of a triangle= $\frac{1}{2}$ of the length of the third side.

  $\begin{array}{l}RV=\sqrt{{\left(13-1\right)}^2+{\left(2-(-4)\right)}^2}\\ =\sqrt{144+36}\\ =\sqrt{180}\\ =6\sqrt{5}\end{array}$

Length of the segment= $\frac{RV}{2}=3\sqrt{5}$