Derive and apply the section formula for internal division of a line segment.
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Fig 7.10 showing similar right triangles PAQ and BPC.
Clean mathematical coordinate graph showing points A(x1, y1), P(x, y), and B(x2, y2) on a line segment. Perpendicular li…
Introduction to finding a point that divides a line segment into specific proportions.
Imagine a telephone company needs to place a relay tower exactly of the way between two existing towns. If they view the towns as points on a map, how do they calculate the exact coordinates for the tower?
The algebraic formula for internal division.
Finds the precise point that divides line segment into the ratio .
Golden Rule: The Criss-Cross
Always multiply the ratio part with the coordinate of the opposite endpoint .
Worked example applying section formula to find coordinates.
Find the coordinates of the point which divides the line segment joining the points and in the ratio internally.
Given:
Guided practice finding point of division.
Let's find the coordinates of the point dividing the line segment joining and in the ratio . Using the section formula, we must remember to criss-cross multiply the ratio parts with the opposite coordinates. For the -coordinate, the setup for the numerator is . The denominator for both coordinates is simply the sum of the ratio parts, which is . Calculating the -coordinate gives . For the -coordinate, applying the exact same pattern gives a numerator of . This results in . Therefore, the required dividing point is .
Order the steps used to derive the section formula.
Arrange the steps to logically derive the section formula from a coordinate plane.