Back to Explain a proof of the Pythagorean Theorem and its converse — Problem 4 · Task Set 26

Exercises: Proving the Pythagorean Theorem and Its Converse

Work through each section in order. The theorem $a^2 + b^2 = c^2$ relates the legs $a$ and $b$ to the hypotenuse $c$ of a right triangle. Show your reasoning where indicated.

Grade 8·22 problems·~35 min·Common Core Math - Grade 8·container·8-g-b-6

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Fluency Practice

Apply the theorem and its converse directly. Show your work.

$A large square with four congruent right triangles in its corners; their hypotenuses form a tilted square in the center. One triangle has legs labeled a and b and hypotenuse labeled c.$

In the area-rearrangement proof, four congruent right triangles are placed in the corners of a large square, leaving a tilted figure in the center. Why does that center figure have four sides of equal length?

Each side of the center figure is the hypotenuse of one of the four congruent triangles, and all four hypotenuses have the same length $c$ .

The four triangles are different sizes, so their hypotenuses average out to equal lengths.

The large square has four equal sides, so the center figure must too.