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In a math test, Emily is asked to find the position of point $M$ that partitions the directed line segment $\overline{KL}$ in the ratio $3:2.$

Solution

The position of a point that partitions a directed line segment in a given ratio can be found by using a dilation. In this case, two right triangles can be drawn to identify the dilation. According to the given ratio, $\overline{KL}$ is divided into $3$ and $2$ parts, represented by $\overline{KM}$ and $\overline{ML}$ respectively, out of $5$ parts in total.

Notice that $\angle KNM$ and $\angle KPL$ are both right angles, and therefore they are congruent angles. Additionally, by the Reflexive Property of Congruence, it can be said that $\angle K$ is congruent to itself.

$$\begin{array}{c}\angle KNM\cong \angle KPL\text{and}\angle K\cong \angle K\end{array}$$

Therefore, by the Angle-Angle Similarity Theorem, it can be concluded that $△KMN$ and $△KLP$ are similar triangles. Furthermore, since $KM$ represents $\frac{3}{5}$ of the length of the directed line segment $\overline{KL},$ it can be said that $△KMN$ is a dilation of $△KLP$ by a scale factor $k$ of $\frac{3}{5}.$

$$\begin{array}{c}k=\frac{3}{5}\end{array}$$

By the composition of the triangles, points $M$ and $N$ are on the same vertical line segment. Therefore, by finding the $x\text{-}$coordinate of $N,$ the $x\text{-}$coordinate of point $M$ can be identified. As derived from the diagram, the length of $\overline{KP}$ is $8$ units. Therefore, $KN$ can be found by multiplying $8$ by the scale factor $k.$

$$\begin{array}{c}KN=\frac{3}{5}\cdot 8\iff KN=4.8\end{array}$$

Thereby, point $N$ is located $4.8$ units away from point $K$ in the positive horizontal direction. As a result, the $x\text{-}$ coordinate of point $N$ is $\text{-}4+4.8=0.8.$ This means that $N$ is located at $(0.8,1).$

Consequently, the $x\text{-}$coordinate of point $M$ is also $0.8.$ To find its $y\text{-}$coordinate, $MN$ should be found. In the diagram, it can be seen that the length of $\overline{LP}$ is $4$ units. With this information, $MN$ can be found by multiplying $4$ by the scale factor.

$$\begin{array}{c}MN=\frac{3}{5}\cdot 4\iff MN=2.4\end{array}$$

Point $M$ is located $2.4$ units away from point $N$ in the positive vertical direction. Since the $y\text{-}$coordinate of $N$ is $1$ and $1+2.4$ is equal to $3.4,$ point $M$ is located at $(0.8,3.4).$

In $△PQR,$ point $S$ partitions $\overline{PQ}$ from point $P$ to point $Q$ in the ratio $3:7.$ Similarly, point $T$ partitions $\overline{RP}$ from point $R$ to point $P$ in the same ratio. With this information, Heichi is trying to find the positions of points $S$ and $T.$

Given the endpoints of the directed line segments, help Heichi determine the coordinates of $S$ and $T.$

Solution

First, the given ratio will be expressed on $\overline{PQ}$ and $\overline{RP}.$

Next, the scale factor $k$ can be found by using the formula for the scale factor of a dilation.

$$\begin{array}{rl}k& =\frac{3}{3+7}\iff k=0.3\end{array}$$

Now, the coordinates of $S$ can be found by substituting the coordinates of the endpoints of $\overline{PQ}$ into the formula for the point that partitions the directed line segment. Note that $S$ divides $\overline{PQ}$ from point $P$ to point $Q.$ Therefore, $P(\text{-}2,5)$ and $Q(\text{-}4,1)$ will be substituted for $(x_1,y_1)$ and $(x_2,y_2),$ respectively.

$S(x_1+k(x_2-x_1),y_1+k(y_2-y_1))$

SubstituteValues

Substitute values

$S(\text{-}2+0.3(\text{-}4-(\text{-}2)),5+0.3(1-5))$

Evaluate

SubNeg

$a-(\text{-}b)=a+b$

$S(\text{-}2+0.3(\text{-}2),5+0.3(1-5))$

SubTerm

Subtract term

$S(\text{-}2+0.3(\text{-}2),5+0.3(\text{-}4))$

MultPosNeg

$a(\text{-}b)=\text{-}a\cdot b$

$S(\text{-}2+(\text{-}0.6),5+(\text{-}1.2))$

AddNeg

$a+(\text{-}b)=a-b$

$S(\text{-}2.6,3.8)$

The coordinates of $S$ were found. By following the same procedure, the coordinates of $T$ can also be found.

Formula: $({\mathbf{x}}_1+\mathbf{k}({\mathbf{x}}_2-{\mathbf{x}}_1),{\mathbf{y}}_1+\mathbf{k}({\mathbf{y}}_2-{\mathbf{y}}_1))$
Segment	Endpoints	Scale Factor	Substitute	Simplify
$\overline{PQ}$	$P(\text{-}2,5)$ and $Q(\text{-}4,1)$	$0.3$	$S(\text{-}2+0.3(\text{-}4-(\text{-}2)),5+0.3(1-5))$	$S(\text{-}2.6,3.8)$
$\overline{RP}$	$R(5,1)$ and $P(\text{-}2,5)$	$0.3$	$T(5+0.3(\text{-}2-5),1+0.3(5-1))$	$T(2.9,2.2)$

Vincenzo and Magdalena are two high school students. They are visiting different cities during their summer holidays. Vincenzo is currently traveling from Madrid to Warsaw, while Magdalena is traveling from Warsaw to Madrid.

The approximated coordinates of the cities are given in the following table.

The distance between the cities is approximately $2000$ kilometers. Give the coordinates of the points at which they would have flown $500$ kilometers. Round the coordinates to the nearest hundredth.

A railroad company is planning to construct a railroad from Houston to Washington D.C. They will place four train stations between the initial and the final station. Each station will be equidistant to each other, as the diagram below shows.

The coordinates of the cities are given in a table.

Find the coordinates of each station. Round them to the nearest hundredth.