Chapter 13.8, Problem 10WE

(a)

To determine

To find: The coordinate of the midpoint $R$ in quadrilateral $SOMK$ .

Answer to Problem 10WE

The coordinate of the midpoint $R$ is $\left(w+t,z\right)$ .

Explanation of Solution

Given information:

Consider the given quadrilateral $SOMK$ .

  

Calculation:

Apply midpoint formula to calculate the coordinates of point $R$ .

  $\begin{array}{c}R=\left(\frac{2w+2t}{2},\frac{2z+0}{2}\right)\\ =\left(\frac{2\left(w+t\right)}{2},\frac{2z}{2}\right)\\ =\left(w+t,z\right)\end{array}$

Conclusion:

Therefore, the coordinate of the midpoint $R$ is $\left(w+t,z\right)$ .

(b)

To determine

To find: The coordinate of the midpoint $E$ in quadrilateral $SOMK$ .

Answer to Problem 10WE

The coordinate of the midpoint $E$ is $\left(u,v\right)$ .

Explanation of Solution

Given information:

Consider the given quadrilateral $SOMK$ .

  

Calculation:

Since, the point $O$ is at the origin. So, the coordinates of the point $O$ is $\left(0,0\right)$ . Apply midpoint formula to calculate the coordinates of point $E$ .

  $\begin{array}{c}E=\left(\frac{2u+0}{2},\frac{2v+0}{2}\right)\\ =\left(\frac{2u}{2},\frac{2v}{2}\right)\\ =\left(u,v\right)\end{array}$

Conclusion:

Therefore, the coordinate of the midpoint $E$ is $\left(u,v\right)$ .

(c)

To determine

To find: The coordinate of the midpoint of the segment $RE$ in quadrilateral $SOMK$ .

Answer to Problem 10WE

The coordinate of the midpoint of the segment $RE$ is $\left(\frac{w+t+u}{2},\frac{z+v}{2}\right)$ .

Explanation of Solution

Given information:

Consider the given quadrilateral $SOMK$ .

  

Calculation:

From part (a), the coordinate of point $R$ is $\left(w+t,z\right)$ . From part (b), the coordinate of point $E$ is $\left(u,v\right)$ . Let the midpoint of the segment is $L$ .

Apply midpoint formula to calculate the coordinates of point $L$ .

  $L=\left(\frac{w+t+u}{2},\frac{z+v}{2}\right)$

Conclusion:

Therefore, the coordinate of the midpoint of the segment $RE$ is $\left(\frac{w+t+u}{2},\frac{z+v}{2}\right)$ .

(d)

To determine

To find: The coordinate of the midpoint $A$ in quadrilateral $SOMK$ .

Answer to Problem 10WE

The coordinate of the midpoint $A$ is $\left(u+w,v+z\right)$ .

Explanation of Solution

Given information:

Consider the given quadrilateral $SOMK$ .

  

Calculation:

Apply midpoint formula to calculate the coordinates of point $A$ .

  $\begin{array}{c}A=\left(\frac{2u+2w}{2},\frac{2v+2z}{2}\right)\\ =\left(\frac{2\left(u+w\right)}{2},\frac{2\left(v+z\right)}{2}\right)\\ =\left(u+w,v+z\right)\end{array}$

Conclusion:

Therefore, the coordinate of the midpoint $A$ is $\left(u+w,v+z\right)$ .

(e)

To determine

To find: The coordinate of the midpoint $H$ in quadrilateral $SOMK$ .

Answer to Problem 10WE

The coordinate of the midpoint $H$ is $\left(t,0\right)$ .

Explanation of Solution

Given information:

Consider the given quadrilateral $SOMK$ .

  

Calculation:

Since, the point $O$ is at the origin. So, the coordinates of the point $O$ is $\left(0,0\right)$ . Apply midpoint formula to calculate the coordinates of point $H$ .

  $\begin{array}{c}H=\left(\frac{0+2t}{2},\frac{0+0}{2}\right)\\ =\left(\frac{2t}{2},0\right)\\ =\left(t,0\right)\end{array}$

Conclusion:

Therefore, the coordinate of the midpoint $H$ is $\left(t,0\right)$ .

(f)

To determine

To find: The coordinate of the midpoint of the segment $AH$ in quadrilateral $SOMK$ .

Answer to Problem 10WE

The coordinate of the midpoint of the segment $AH$ is $\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)$ .

Explanation of Solution

Given information:

Consider the given quadrilateral $SOMK$ .

  

Calculation:

From part (d), the coordinate of point $A$ is $\left(u+w,v+z\right)$ . From part (e), the coordinate of point $H$ is $\left(t,0\right)$ . Let the midpoint of the segment is $N$ .

Apply midpoint formula to calculate the coordinates of point $N$ .

  $\begin{array}{c}N=\left(\frac{u+w+t}{2},\frac{v+z+0}{2}\right)\\ =\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)\end{array}$

Conclusion:

Therefore, the coordinate of the midpoint of the segment $AH$ is $\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)$ .

(g)

To determine

To find: The coordinate of the intersection of the segment $RE$ and the segment $AH$ .

Answer to Problem 10WE

The coordinate of the intersection of the segment $RE$ and the segment $AH$ is $\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)$ .

Explanation of Solution

Given information:

Consider the given quadrilateral $SOMK$ .

  

Calculation:

From part (c), the coordinate of the midpoint of the segment $RE$ is $\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)$ . From part (f), the coordinate of the midpoint of the segment $AH$ is $\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)$ .

Since, the segment $RE$ and the segment $AH$ bisects each other a common point which is the midpoint of the each segment. So, the coordinate of the intersection of the segment $RE$ and the segment $AH$ is $\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)$ .

Conclusion:

Therefore, the coordinate of the intersection of the segment $RE$ and the segment $AH$ is $\left(\frac{u+w+t}{2},\frac{v+z}{2}\right)$ .