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A point on a directed line segment partitions the segment in a specific ratio. In this lesson, given the segment's endpoints and ratio, the point's coordinates will be identified.
### Catch-Up and Review

Given that vertices $A,$ $B,$ and $C$ are collinear and $BE$ is parallel to $CD,$ choose the correct similarity theorem that proves $△AEB∼△BDC.$
Given that $AB=5,$ $AE=4,$ and $CD=6,$ identify the scale factor of $△ABE$ to $△ACD$ and find the length of $AC.$
If the hypotenuse of $△ADE$ is $4,$ find the hypotenuse of $△ABC.$

**Here are a few recommended readings before getting started with this lesson.**

Try your knowledge on these topics.

a Consider the given right triangles, with two parallel legs.

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b In the following diagram, $△ABE$ is similar to $△ACD.$

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c Given two right triangles, $△ABC$ is the dilation of $△ADE$ by a scale factor of $25 .$

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Challenge

Dylan's home is $4.5$ miles away from Big Apple Circus. Dylan walks from his home to the circus to watch the flying trapeze performance.

What are the coordinates of the point at which Dylan would have walked $3$ miles?

Example

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

Help her identify the coordinates of $M.${"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"decimal":false,"function":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.10903em;\">M<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text1":"0.8","text2":"3.4"}}

Draw a right triangle with hypotenuse $KM$ and another right triangle with hypotenuse $KL.$

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, $KL$ is divided into $3$ and $2$ parts, represented by $KM$ and $ML$ respectively, out of $5$ parts in total.

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

Therefore, by the Angle-Angle Similarity Theorem, it can be concluded that $△KMN$ and $△KLP$ are similar triangles. Furthermore, since $KM$ represents $53 $ of the length of the directed line segment $KL,$ it can be said that $△KMN$ is a dilation of $△KLP$ by a scale factor $k$ of $53 .$
$k=53 $

By the composition of the triangles, points $M$ and $N$ are on the same vertical line segment. Therefore, by finding the $x-$coordinate of $N,$ the $x-$coordinate of point $M$ can be identified. As derived from the diagram, the length of $KP$ is $8$ units. Therefore, $KN$ can be found by multiplying $8$ by the scale factor $k.$
$KN=53 ⋅8⇔KN=4.8 $

Thereby, point $N$ is located $4.8$ units away from point $K$ in the positive horizontal direction. As a result, the $x-$ coordinate of point $N$ is $-4+4.8=0.8.$ This means that $N$ is located at $(0.8,1).$
Consequently, the $x-$coordinate of point $M$ is also $0.8.$ To find its $y-$coordinate, $MN$ should be found. In the diagram, it can be seen that the length of $LP$ is $4$ units. With this information, $MN$ can be found by multiplying $4$ by the scale factor.
$MN=53 ⋅4⇔MN=2.4 $

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

The approach used to find the coordinates of point $M$ can be generalized algebraically. Suppose that $M$ divides the directed line segment from $A(x_{1},y_{1})$ to $B(x_{2},y_{2})$ in the ratio $a:b.$

Since $M$ divides $AB$ from $A$ to $B,$ point $A$ is said to be the center of dilation. Next, two right triangles will be considered to show the dilation. The directed distances from $A$ to $B$ in the $x-$ and $y-$directions can be identified by subtracting the endpoints of the segment.

Since $∠ADM$ and $∠ACB$ are right angles, they are congruent to each other. Furthermore, by the Reflexive Property of Congruence, it can be stated $∠A$ is congruent to itself. With this information, by the Angle-Angle Similarity Theorem, $△AMD$ and $△ABC$ are similar triangles.

$△AMD∼△ABC $

The next step to identify the position of $M$ is to express the ratio $a:b$ as the scale factor of the dilation. Since the hypotenuses of $△AMD$ and $△ABC$ are $a$ and $a+b,$ respectively, the scale factor $k$ is the quotient between $a$ and $a+b.$ $k=a+ba $

Because $△AMD$ and $△ABC$ are similar triangles, $AD$ and $MD$ can be found by multiplying $AC$ and $BC,$ respectively, by the scale factor $k.$
$AD=kAC⇓AD=k(x_{2}−x_{1}) MD=kBC⇓MD=k(y_{2}−y_{1}) $

Finally, the coordinates of $M$ can be found by adding the resulting distances to the coordinates of the center of dilation. $M(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

$⇕$

$M(x_{1}+a+ba (x_{2}−x_{1}),y_{1}+a+ba (y_{2}−y_{1}))$

Example

In $△PQR,$ point $S$ partitions $PQ $ from point $P$ to point $Q$ in the ratio $3:7.$ Similarly, point $T$ partitions $RP$ 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 $T.$

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Use the formula for identifying the position of a point on a directed line segment.

First, the given ratio will be expressed on $PQ $ and $RP.$

Next, the scale factor $k$ can be found by using the formula for the scale factor of a dilation.$k =3+73 ⇔k=0.3 $

Now, the coordinates of $S$ can be found by substituting the coordinates of the endpoints of $PQ $ into the formula for the point that partitions the directed line segment. Note that $S$ divides $PQ $ from point $P$ to point $Q.$ Therefore, $P(-2,5)$ and $Q(-4,1)$ 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

$S(-2+0.3(-4−(-2)),5+0.3(1−5))$

Evaluate

SubNeg

$a−(-b)=a+b$

$S(-2+0.3(-2),5+0.3(1−5))$

SubTerm

Subtract term

$S(-2+0.3(-2),5+0.3(-4))$

MultPosNeg

$a(-b)=-a⋅b$

$S(-2+(-0.6),5+(-1.2))$

AddNeg

$a+(-b)=a−b$

$S(-2.6,3.8)$

Formula: $(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$ | ||||
---|---|---|---|---|

Segment | Endpoints | Scale Factor | Substitute | Simplify |

$PQ $ | $P(-2,5)$ and $Q(-4,1)$ | $0.3$ | $S(-2+0.3(-4−(-2)),5+0.3(1−5))$ | $S(-2.6,3.8)$ |

$RP$ | $R(5,1)$ and $P(-2,5)$ | $0.3$ | $T(5+0.3(-2−5),1+0.3(5−1))$ | $T(2.9,2.2)$ |

Example

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 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. ### Hint

### Solution

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

City | Longitude ($x$) | Latitude ($y$) |
---|---|---|

Madrid | $≈-3.70$ | $≈40.42$ |

Warsaw | $≈21.01$ | $≈52.23$ |

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Use the formula for identifying the position of a point on a directed line segment.

The coordinates of points $V$ and $M$ can be found by using the formula for identifying the position of a point on a directed line segment. The distance between Madrid and Warsaw is $2000$ kilometers. It is given that each passenger has flown $500$ kilometers. With this information, the scale factor of the dilation can be identified.
By following the same procedure, the coordinates of $M$ can also be found. Be aware that Magdalena is traveling in the opposite direction of Vincenzo, from Warsaw to Madrid, while Vincenzo is traveling from Madrid to Warsaw.

$k=2000500 ⇔k=0.25 $

The scale factor for both routes is $0.25.$ From here, the coordinates of $V$ can be found by substituting the coordinates of the cities and the scale factor into the formula. Since Vincenzo is traveling from Madrid to Warsaw, $(-3.70,40.42)$ and $(21.01,52.23)$ will be substituted for $(x_{1},y_{1})$ and $(x_{2},y_{2}),$ respectively.
$V(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

SubstituteValues

Substitute values

$V(-3.70+0.25(21.01−(-3.70)),40.42+0.25(52.23−40.42))$

$V(2.48,43.37)$

Formula: $(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$ | ||||
---|---|---|---|---|

Route | Endpoints | Scale Factor | Substitute | Simplify |

Madrid to Warsaw | $(-3.70,40.42)$ and $(21.01,52.23)$ | $0.25$ | $V(-3.70+0.25(21.01−(-3.70)),40.42+0.25(52.23−40.42))$ | $V(2.48,43.37)$ |

Warsaw to Madrid | $(21.01,52.23)$ and $(-3.70,40.42)$ | $0.25$ | $M(21.01+0.25(-3.70−21.01),52.23+0.25(40.42−52.23))$ | $M(14.83,49.28)$ |

Even if Vincenzo and Magdalena travel between the same cities, they are in different positions. The reason for their differing positions is because they are traveling in opposite directions. This reasoning would suggest, a point's position on a directed line segment depends on which endpoint is used as the center of dilation.

Example

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.

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

### Solution

The coordinates of the cities are given in a table.

City | Longitude ($x$) | Latitude ($y$) |
---|---|---|

Houston | $≈-95.37$ | $≈29.76$ |

Washington D.C. | $≈-77.04$ | $≈38.91$ |

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Use the formula for identifying the position of a point on a directed line segment.

Consider station $A.$ Because each station is equidistant — the same distance to eachother — station $A$ partitions the railroad from Houston to Washington in the ratio $1:4.$ By using this information, the scale factor of the dilation can be found.
By following the same procedure, the coordinates of the other stations can also be found. First, the scale factors should be identified for each station.

$k=1+41 ⇔k=0.2 $

Now, by substituting the scale factor and the coordinates of the cities into the formula, the coordinates of station $A$ can be found.
$A(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

SubstituteValues

Substitute values

$A(-95.37+0.2(-77.04−(-95.37)),29.76+0.2(38.91−29.76))$

$A(-91.70,31.59)$

Station | Ratio | Scale Factor | Simplify |
---|---|---|---|

$A$ | $1:4$ | $k=1+41 $ | $k=0.2$ |

$B$ | $2:3$ | $k=2+32 $ | $k=0.4$ |

$C$ | $3:2$ | $k=3+23 $ | $k=0.6$ |

$D$ | $4:1$ | $k=4+14 $ | $k=0.8$ |

From here, the positions of the other stations can be identified. Keep in mind that all stations go from Houston to Washington. Therefore, the endpoints are always $(-95.37,29.76)$ and $(-77.04,38.91).$

Formula: $(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$ | |||
---|---|---|---|

Station | Scale Factor | Substitute | Simplify |

$A$ | $0.2$ | $A(-95.37+0.2(-77.04−(-95.37)),29.76+0.2(38.91−29.76))$ | $A(-91.70,31.59)$ |

$B$ | $0.4$ | $B(-95.37+0.4(-77.04−(-95.37)),29.76+0.4(38.91−29.76))$ | $B(-88.04,33.42)$ |

$C$ | $0.6$ | $C(-95.37+0.6(-77.04−(-95.37)),29.76+0.6(38.91−29.76))$ | $C(-84.37,35.25)$ |

$D$ | $0.8$ | $D(-95.37+0.8(-77.04−(-95.37)),29.76+0.8(38.91−29.76))$ | $D(-80.71,37.08)$ |

Closure

By using the formula learned in this lesson, the challenge presented at the beginning can be solved. Recall that the coordinates of the point at which Dylan would have walked $3$ miles of the $4.5-$mile distance were asked to be found.

### Hint

### Solution

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Determine the scale factor of the dilation. Use the formula for identifying the position of a point on a directed line segment.

Notice that the point at which Dylan stands after he walked $3$ miles partitions the road in the ratio $2:1.$
Therefore, the coordinates of the point at which Dylan would have walked $3$ miles are $(5,3).$

$1.53 ⇔2:1 $

With this information, the scale factor of the dilation can be determined.
$k=2+12 ⇔k=32 $

By substituting the scale factor and the coordinates of the endpoints into the corresponding formula, Dylan's position can be found. Since he walks from his house to the circus, $(2,5)$ and $(6.5,2)$ will be substituted for $(x_{1},y_{1})$ and $(x_{2},y_{2}),$ respectively.
$D(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

SubstituteValues

Substitute values

$D(2+32 (6.5−2),5+32 (2−5))$

Evaluate

SubTerms

Subtract terms

$D(2+32 (4.5),5+32 (-3))$

MultPosNeg

$a(-b)=-a⋅b$

$D(2+32 (4.5),5−32 (3))$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$D(2+39 ,5−36 )$

CalcQuot

Calculate quotient

$D(2+3,5−2)$

AddSubTerms

Add and subtract terms

$D(5,3)$