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Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
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.
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.
M(x1+k(x2−x1),y1+k(y2−y1))
⇕
M(x1+a+ba(x2−x1),y1+a+ba(y2−y1))
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.
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.
Substitute values
a−(-b)=a+b
Subtract term
a(-b)=-a⋅b
a+(-b)=a−b
Formula: (x1+k(x2−x1),y1+k(y2−y1)) | ||||
---|---|---|---|---|
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.
Use the formula for identifying the position of a point on a directed line segment.
Substitute values
Formula: (x1+k(x2−x1),y1+k(y2−y1)) | ||||
---|---|---|---|---|
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.
Use the formula for identifying the position of a point on a directed line segment.
Substitute values
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: (x1+k(x2−x1),y1+k(y2−y1)) | |||
---|---|---|---|
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) |
Determine the scale factor of the dilation. Use the formula for identifying the position of a point on a directed line segment.
Substitute values
Subtract terms
a(-b)=-a⋅b
ca⋅b=ca⋅b
Calculate quotient
Add and subtract terms