Applying the Pythagorean theorem

Japanese school year: Junior high 3

What you learn

Apply the Pythagorean theorem to find distances in coordinate planes and calculate lengths in three-dimensional shapes. This skill is widely used in advanced geometry, physics, and computer graphics. Prior experience in calculating side lengths of right triangles using the theorem is recommended.

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Key points

This formula finds the distance $d$ between two points on a coordinate plane. Square the differences in the $x$ and $y$ coordinates, add them, and take the square root.

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

This formula calculates the diagonal length $l$ inside a rectangular prism. Square the length $a$ , width $b$ , and height $c$ , add them together, and take the square root.

l = \sqrt{a^2 + b^2 + c^2}

This formula calculates the distance $d$ between two points in 3D space. Square the differences along the $x$ , $y$ , and $z$ axes, add them, and take the square root.

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

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Practice 1Coming soon