Length and midpoint.
Use Pythagoras for distance and coordinate averages for the halfway point.
Calculate exact and decimal lengths · find midpoints · recover a missing endpoint
Build a right-angled triangle
The distance between two points is the length of the straight segment joining them. When the points are not horizontally or vertically aligned, use their coordinate changes to form the legs of a right-angled triangle.
Pythagoras connects those horizontal and vertical legs to the diagonal segment. Square each difference, add the squares and take the non-negative square root. Negative coordinate changes cause no problem because length depends on their squared sizes, not the travel direction.
The horizontal and vertical differences form the legs of a right-angled triangle. Pythagoras gives the direct distance between its endpoints.
Find the length from (−2, −1) to (4, 3).
Leave the surd for an exact answer, or give about 7.21 units if a decimal approximation is requested.
Pause and explainWhy should you not simply add the horizontal and vertical gaps?
That gives the length of an L-shaped route. The straight diagonal is a different, shorter distance.
Midpoint means average each coordinate
A midpoint sits halfway along a segment. Its horizontal position is halfway between the endpoint x-values, and its vertical position is halfway between the endpoint y-values.
Average each coordinate separately. If an endpoint is missing but the midpoint is known, work backward: the full coordinate gap extends equally far on the other side of the midpoint. This is why doubling the midpoint coordinate and subtracting the known endpoint recovers the other endpoint.
Find the midpoint of A (−2, −1) and B (4, 3).
Check: M is 3 units right and 2 units up from A, exactly half of the full changes.
M (2, −1) is the midpoint of A (−3, 4) and B. Find B.
If B = (x, y), then (−3 + x) ÷ 2 = 2 and (4 + y) ÷ 2 = −1.
Pause and explainWhat is the midpoint between x-values −2 and 6?
2, since (−2 + 6)2 = 2. The same averaging idea applies separately to y.
Your turn. Keep length and midpoint distinct.
Read whether the question asks for a distance, a midpoint or an endpoint. A distance is one non-negative number; a midpoint or endpoint is a coordinate pair.
For distance, show the two coordinate differences. For a midpoint, average matching coordinates. Use a quick sketch to check that a midpoint lies between the endpoints.
Find the length and midpoint of the segment from (−1, 2) to (5, 10).
Reveal worked answer
Find the exact distance between (2, −3) and (−1, 1).
Reveal worked answer
The midpoint of A (1, 5) and B is (4, 2). Find B.