Question Video: Using the Ratio of Division of a Line Segment to Find the Length of Another Line Segment in a Coordinate System | Nagwa Question Video: Using the Ratio of Division of a Line Segment to Find the Length of Another Line Segment in a Coordinate System | Nagwa

Question Video: Using the Ratio of Division of a Line Segment to Find the Length of Another Line Segment in a Coordinate System Mathematics • First Year of Secondary School

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Points 𝐴, 𝐡, 𝐢, and 𝐷 have coordinates (βˆ’15, βˆ’7), (7, 2), (4, βˆ’17), and (13, βˆ’2), respectively. 𝐸 is the midpoint of the line segment 𝐴𝐡, and 𝑀 divides 𝐢𝐷 externally by the ratio 7 : 4. Find the length of the line segment 𝐸𝑀 to the nearest hundredth, considering a length unit = 1 cm.

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Video Transcript

Points 𝐴, 𝐡, 𝐢, and 𝐷 have coordinates negative 15, negative seven; seven, two; four, negative 17; and 13, negative two, respectively. 𝐸 is the midpoint of the line segment 𝐴𝐡, and 𝑀 divides 𝐢𝐷 externally by the ratio seven to four. Find the length of the line segment 𝐸𝑀 to the nearest hundredth, considering a length unit equals one centimeter.

To find the length of the line segment 𝐸𝑀, we must first determine the coordinates of points 𝐸 and 𝑀. Let’s consider point 𝐸 first of all. We’re told that point 𝐸 is the midpoint of the line segment 𝐴𝐡. We can recall that the midpoint of the line segment connecting the points π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two is π‘₯ one plus π‘₯ two over two, 𝑦 one plus 𝑦 two over two. In other words, the π‘₯-coordinate of the midpoint is the average of the π‘₯-coordinates of the two endpoints of the line segment. And the 𝑦-coordinate of the midpoint is the average of the 𝑦-coordinates of the two endpoints.

So, given that 𝐴 has coordinates negative 15, negative seven and 𝐡 has coordinates seven, two, the π‘₯-coordinate of point 𝐸 is negative 15 plus seven over two. And the 𝑦-coordinate of point 𝐸 is negative seven plus two over two. That simplifies to negative eight over two, negative five over two, which we can write as the point negative four, negative 2.5. So we determined the coordinates of point 𝐸.

Next, we need to determine the coordinates of point 𝑀. We’re told that 𝑀 divides the line 𝐢𝐷 externally in the ratio seven to four. This means that 𝑀 is on the extension of the line connecting point 𝐢 to point 𝐷. And the ratio of the length of the line segment 𝐢𝑀 to the length of the line segment 𝐷𝑀 is seven to four. We can work out the coordinates of point 𝑀 in two ways, either formally using the section formula with external division or more practically.

Let’s consider the section formula first. This states that if we have distinct points 𝐴 π‘₯ one, 𝑦 one and 𝐡 π‘₯ two, 𝑦 two and point 𝑃, which does not lie on the line segment 𝐴𝐡, divides the line 𝐴𝐡 such that the ratio of 𝐴𝑃 to 𝐡𝑃 is π‘š to 𝑛, then 𝑃 has coordinates π‘šπ‘₯ two minus 𝑛π‘₯ one over π‘š minus 𝑛, π‘šπ‘¦ two minus 𝑛𝑦 one over π‘š minus 𝑛. We may find it helpful to swap the letters here for 𝐢, 𝐷, and 𝑀. So we’re told that if the ratio of 𝐢𝑀 to 𝐷𝑀 is π‘š to 𝑛, and we know that the ratio of 𝐢𝑀 to 𝐷𝑀 is seven to four.

We also know that the coordinates of point 𝐢, which are going to be π‘₯ one, 𝑦 one, are four, negative 17 and the coordinates of point 𝐷, which are going to be π‘₯ two, 𝑦 two, are 13, negative two. So, substituting these values, we have that the coordinates of point 𝑀 are seven multiplied by 13 minus four multiplied by four over seven minus four and seven multiplied by negative two minus four multiplied by negative 17 over seven minus four. That simplifies to 75 over three, 54 over three, which further simplifies to give the point 25, 18.

So that was the formal method using the section formula with external division. But we can also think about this problem from a practical perspective. If the ratio of the length of 𝐢𝑀 to 𝐷𝑀 is seven to four, then the ratio of the length of 𝐢𝐷 to 𝐷𝑀 is three to four. To move from 𝐢 to 𝐷, the π‘₯-coordinate increases by nine and the 𝑦-coordinate increases by 15. So, to get from 𝐷 to 𝑀, the π‘₯-coordinate will need to increase by four-thirds of nine, which is 12, and the 𝑦-coordinate will need to increase by four-thirds of 15, which is 20. So an alternative way to work out the coordinates of point 𝑀 would be to add 12 to the π‘₯-coordinate of point 𝐷, 13 plus 12, which gives 25, and to add 20 to the 𝑦-coordinate of point 𝐷, negative two plus 20, which gives 18.

Now that we’ve found the coordinates of points 𝐸 and 𝑀, we need to calculate the length of the line segment 𝐸𝑀, which we can do using the distance formula. This tells us that the distance between the two points with coordinates π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two is equal to the square root of π‘₯ two minus π‘₯ one squared plus 𝑦 two minus 𝑦 one squared. So, substituting the coordinates of 𝐸 and 𝑀, we have the square root of 25 minus negative four squared plus 18 minus negative 2.5 squared. That’s the square root of 29 squared plus 20.5 squared, which evaluates to 35.5140 continuing.

We’re asked to give the length to the nearest hundredth, considering a length unit is one centimeter. So we found that the length of the line segment 𝐸𝑀 to the nearest hundredth is 35.51 centimeters.

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