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5.B: Pythagorean Theorem and Distance

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)


    The purpose of this lesson is to apply the Pythagorean Theorem to various situations in two and three dimensions.

    This lesson will address the following CCRS Standard(s) for Geometry:

    • 8.G.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions


    1. Take notes while watching videos below
    2. Go to and log into our course to complete assignment 5.C with 80% or better.


    Complete assignment 5.C with 80% or better at


    In this lesson we have learned:

    • In two dimensions, the Pythagorean Theorem is LaTeX: a^2+b^2=c^2
    • In three dimensions, the Pythagorean Theorem is LaTeX: a^2+b^2+c^2=d^2
    • The distance between two points can be found using the Pythagorean Theorem. 
      • The distance between LaTeX: \left(x_1,y_1\right) and LaTeX: \left(x_2,\:y_2\right) is LaTeX: d=\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>&#x2212;</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </msqrt> </math>' data-equation-content="d=\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}">

    5.B: Pythagorean Theorem and Distance is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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