Distance Formula Math Example 2

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Example 2

medium
Find the distance between (−3,5)(-3, 5) and (2,−7)(2, -7).

Solution

  1. 1
    Apply d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} with (x1,y1)=(−3,5)(x_1, y_1) = (-3, 5) and (x2,y2)=(2,−7)(x_2, y_2) = (2, -7).
  2. 2
    Compute the differences carefully, watching the signs: x2−x1=2−(−3)=5x_2 - x_1 = 2 - (-3) = 5 and y2−y1=−7−5=−12y_2 - y_1 = -7 - 5 = -12. Square each: 52=255^2 = 25, (−12)2=144(-12)^2 = 144.
  3. 3
    Substitute: d=25+144=169=13d = \sqrt{25 + 144} = \sqrt{169} = 13. Recognise the 5-12-13 Pythagorean triple.

Answer

d=13d = 13
When coordinates include negatives, be careful with signs. Subtracting a negative is addition: 2−(−3)=52 - (-3) = 5. The squaring ensures the distance is always positive regardless of direction.

About Distance Formula

A formula for finding the distance between two points in the coordinate plane, derived directly from the Pythagorean theorem.

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More Distance Formula Examples

Example 1 easy

Find the distance between the points [formula] and [formula].

Example 3 medium

Show that the triangle with vertices [formula], [formula], and [formula] is isosceles.

Example 4 medium

Find the points on the [formula]-axis that are [formula] units away from the point [formula].

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