Task Sets: Explain a proof of the Pythagorean Theorem and its converse - Common Core Math - Grade 8

Recall / Warm-Up

These problems review skills you already know: parts of a right triangle, squares of numbers, and triangle angle sums.

$A right triangle with the right angle at the bottom-left, sides labeled p (vertical leg), q (horizontal leg), and r (slanted side opposite the right angle).$

In the right triangle shown, which side is the hypotenuse?

Side $r$ (the side opposite the right angle)

Side $p$

Side $q$

The two acute angles of a right triangle measure $37\degree$ and $x$ . What is $x$ ?

$53\degree$

$143\degree$

$63\degree$

What does it mean for one statement to be the "converse" of another? Give one everyday (non-math) example of an "if-then" statement and its converse.

Fluency Practice

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

A right triangle has legs of length 3 and 4. Find the length of the hypotenuse $c$ .

A triangle has sides 6, 8, and 10. Compute $6^2 + 8^2$ and compare it to $10^2$ to decide whether the triangle is a right triangle. Enter "yes" or "no".

To test whether a triangle with sides 9, 12, and 15 is a right triangle, which equation should you check?

$9^2 + 12^2 = 15^2$

$9^2 + 15^2 = 12^2$

$12^2 + 15^2 = 9^2$

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

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.

A triangle has sides 5, 7, and 9. The longest side is 9. Compute $5^2 + 7^2$ .

Exercises: Proving the Pythagorean Theorem and Its Converse

Recall / Warm-Up

Fluency Practice