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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.**

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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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?

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=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"variable\">M<\/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 $53 .$
The approach used to find the coordinates of point M can be generalized algebraically. Suppose that M divides the directed line segment from A(x1,y1) to B(x2,y2) in the ratio a:b.
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.
Because △AMD and △ABC are similar triangles, AD and MD can be found by multiplying AC and BC, respectively, by the scale factor k.
Finally, the coordinates of M can be found by adding the resulting distances to the coordinates of the center of dilation.

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.

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.$S(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

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) |

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.

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

Madrid | $≈-3.70$ | $≈40.42$ |

Warsaw | $≈21.01$ | $≈52.23$ |

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.

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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.
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 (x1,y1) and (x2,y2), respectively.
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.

$V(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

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.

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.

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

Houston | $≈-95.37$ | $≈29.76$ |

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

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

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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.
Now, by substituting the scale factor and the coordinates of the cities into the formula, the coordinates of station A 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.

$A(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

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) |

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.
With this information, the scale factor of the dilation can be determined.
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 (x1,y1) and (x2,y2), respectively.
Therefore, the coordinates of the point at which Dylan would have walked 3 miles are (5,3).

$D(x_{1}+k(x_{2}−x_{1}),y_{1}+k(y_{2}−y_{1}))$

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)

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