Derive and apply the distance formula to find the length of segments on the coordinate plane.
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Hook explaining the need to calculate distances not parallel to the axes.
When points are aligned horizontally or vertically, finding the distance is simple: we just count the units along the grid or subtract the values on that axis.
Visual of triangle ACD used for distance derivation.
Creating a right-angled triangle ACD helps us calculate the slanted length of side AD.
Step-by-step calculation using coordinates and Pythagoras theorem.
Problem. Find the length of side AD for triangle ADM, given the starting coordinate and the ending coordinate .
Derived from the Baudhāyana-Pythagoras Theorem, this formula calculates the straight-line distance between any two points in the 2-D plane.
After adding the squares of the coordinate differences, you must remember to take the square root at the very end to find the actual distance.
Faded example calculating the length of side DM.
We want to find the distance between A(-4, 6) and B(5, -2). The distance formula is . First think about how far the x-coordinate moves from A to B. The change in x-coordinate is . Next think about how far the y-coordinate moves from A to B. The change in y-coordinate is . Now use these two changes in the distance formula. After squaring both changes and adding them, the number under the square root becomes . Therefore, the distance between the two points is units.
Exploring how reflections affect coordinate signs but preserve distance.
What happens to the lengths of sides if we reflect a shape across one of the axes? Let's take triangle ADM and reflect it across the y-axis to create a mirror image, triangle A'D'M'.