Chapter 14.6, Problem 11WE

a.

To determine

To find $H_{N}o{H}_M:S\to \underset{---}{?}$

Answer to Problem 11WE

The value of $H_{N}o{H}_M:S\to \text{ is }Q$ .

Explanation of Solution

Given triangle $QRS$ and $L,M,\text{and }N$ are the midpoints of the sides of the triangle $QRS$ .

Calculation:

Consider a composite of two half-turn transformations; $H_{N}o{H}_M$

It states that first half-turn rotation with respect to the point $M$ and then will respect to the point $N$ .

  $H_M:S\to R\text{ and }{H}_N:R\to Q.$

Thus, the value of $H_{N}o{H}_M:S\to \text{ is }Q.$

b.

To determine

To find $H_{M}o{H}_N:Q\to \underset{---}{?}$

Answer to Problem 11WE

The value of $H_{M}o{H}_N:Q\to \text{ is }S.$

Explanation of Solution

Given triangle $QRS$ and $L,M,\text{and }N$ are the midpoints of the sides of the triangle $QRS$ .

Calculation:: Consider a composite of two half-turn transformations; $H_{M}o{H}_N.$

It states that first half-turn rotation with respect to the point $N$ and then will respect to the point $M$ .

  $H_N:Q\to R,\text{and},H_M:R\to S$

Thus, the value of $H_{M}o{H}_N:Q\to \text{ is }S.$

c.

To determine

To find $D_{S,\frac{1}{2}}o{H}_N:Q\to \underset{---}{?}$

Answer to Problem 11WE

The value of $D_{S,\frac{1}{2}}o{H}_N:Q\to \text{ is }M.$

Explanation of Solution

Given triangle $QRS$ and $L,M,\text{and }N$ are the midpoints of the sides of the triangle $QRS$ .

Calculation:: Consider a composite of two half-turn transformations; $D_{S\cdot \frac{1}{2}}o{H}_N$

It states that first half-turn rotation with respect to the point $N$ and then a dilation of a scalar factor $\frac{1}{2}$ with respect to the point $S$ .

  $H_N:Q\to R,\text{and},D_{S,\frac{1}{2}}:R\to M.$

Thus, the value of $D_{S,\frac{1}{2}}o{H}_N:Q\to \text{ is }M.$

d.

To determine

To find $D_{S,2}o{H}_N:M\to \underset{---}{?}$

Answer to Problem 11WE

The value of $D_{S,2}o{H}_N:M\to \text{ is }Q.$

Explanation of Solution

Given triangle $QRS$ and $L,M,\text{and }N$ are the midpoints of the sides of the triangle $QRS$ .

Calculation:

Consider a composite of two half-turn transformations; $H_{N}o{D}_{S,2}$

It states that adilation of scalar factor $2$ will respect to the point $S$ and then half-turn rotation with respect to the point $N$

  $D_{S,2}:M\to R,\text{and},H_N:R\to Q.$

Thus, the value of $D_{S,2}o{H}_N:M\to \text{ is }Q.$

e.

To determine

To find $H_{L}o{H}_{M}o{H}_N:Q\to \underset{---}{?}$

Answer to Problem 11WE

The value of $H_{L}o{H}_{M}o{H}_N:Q\to \text{ is }Q.$

Explanation of Solution

Given triangle $QRS$ and $L,M,\text{and }N$ are the midpoints of the sides of the triangle $QRS$ .

Calculation:

Consider a composite of two half-turn transformations; $H_{L}o{H}_{M}o{H}_N$

It states that first half-turn rotation with respect to the point $N$ and then will respect to the point $M$ and half turn will respect to the point $L$ .

  $H_N:Q\to R,\text{and},H_M:R\to S.$

  $\text{Similarly},H_L:S\to Q.$

Thus, the value of $H_{L}o{H}_{M}o{H}_N:Q\to \text{ is }Q.$