### Video Transcript

Points π΄, π΅, πΆ, and π· have coordinates negative 15, negative seven; seven, two; four, negative 17; and 13, negative two, respectively. πΈ is the midpoint of the line segment π΄π΅, and π divides πΆπ· externally by the ratio seven to four. Find the length of the line segment πΈπ to the nearest hundredth, considering a length unit equals one centimeter.

To find the length of the line segment πΈπ, we must first determine the coordinates of points πΈ and π. Letβs consider point πΈ first of all. Weβre told that point πΈ is the midpoint of the line segment π΄π΅. We can recall that the midpoint of the line segment connecting the points π₯ one, π¦ one and π₯ two, π¦ two is π₯ one plus π₯ two over two, π¦ one plus π¦ 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 π¦-coordinate of the midpoint is the average of the π¦-coordinates of the two endpoints.

So, given that π΄ has coordinates negative 15, negative seven and π΅ has coordinates seven, two, the π₯-coordinate of point πΈ is negative 15 plus seven over two. And the π¦-coordinate of point πΈ 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 πΈ.

Next, we need to determine the coordinates of point π. Weβre told that π divides the line πΆπ· externally in the ratio seven to four. This means that π is on the extension of the line connecting point πΆ to point π·. And the ratio of the length of the line segment πΆπ to the length of the line segment π·π is seven to four. We can work out the coordinates of point π in two ways, either formally using the section formula with external division or more practically.

Letβs consider the section formula first. This states that if we have distinct points π΄ π₯ one, π¦ one and π΅ π₯ two, π¦ two and point π, which does not lie on the line segment π΄π΅, divides the line π΄π΅ such that the ratio of π΄π to π΅π is π to π, then π has coordinates ππ₯ two minus ππ₯ one over π minus π, ππ¦ two minus ππ¦ one over π minus π. We may find it helpful to swap the letters here for πΆ, π·, and π. So weβre told that if the ratio of πΆπ to π·π is π to π, and we know that the ratio of πΆπ to π·π is seven to four.

We also know that the coordinates of point πΆ, which are going to be π₯ one, π¦ one, are four, negative 17 and the coordinates of point π·, which are going to be π₯ two, π¦ two, are 13, negative two. So, substituting these values, we have that the coordinates of point π 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 πΆπ to π·π is seven to four, then the ratio of the length of πΆπ· to π·π is three to four. To move from πΆ to π·, the π₯-coordinate increases by nine and the π¦-coordinate increases by 15. So, to get from π· to π, the π₯-coordinate will need to increase by four-thirds of nine, which is 12, and the π¦-coordinate will need to increase by four-thirds of 15, which is 20. So an alternative way to work out the coordinates of point π would be to add 12 to the π₯-coordinate of point π·, 13 plus 12, which gives 25, and to add 20 to the π¦-coordinate of point π·, negative two plus 20, which gives 18.

Now that weβve found the coordinates of points πΈ and π, we need to calculate the length of the line segment πΈπ, which we can do using the distance formula. This tells us that the distance between the two points with coordinates π₯ one, π¦ one and π₯ two, π¦ two is equal to the square root of π₯ two minus π₯ one squared plus π¦ two minus π¦ one squared. So, substituting the coordinates of πΈ and π, we have the square root of 25 minus negative four squared plus 18 minus negative 2.5 squared. Thatβs the square root of 29 squared plus 20.5 squared, which evaluates to 35.5140 continuing.

Weβre 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 πΈπ to the nearest hundredth is 35.51 centimeters.