Update all PDFs

# Applying the Pythagorean Theorem in a mathematical context

Alignments to Content Standards: 8.G.B

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure below shows some of the dimensions but is not drawn to scale.

## Solutions

Using the Pythagorean Theorem, we can find the hypotenuse $c$ of the smallest right triangle.

$$5^2 + 5^2 = c^2$$

so

$$c = \sqrt{5^2+5^2} = 5 \sqrt{2}.$$

Similarly, the hypotenuse of the right triangle with side lengths 7 and 14 is

$$\sqrt{7^2+14^2} = 7 \sqrt{5}.$$

If the shaded triangle is a right triangle, then the side-lengths must satisfy the Pythagorean Theorem. Since $7 \sqrt{5}$ is the longest of the three sides, it would be the hypotenuse, so if this is a right triangle, then the following equation must be true:

$$(5 \sqrt{2})^2 + 15^2 = (7 \sqrt{5})^2.$$

However, looking at the left-hand side, we find that

$$(5 \sqrt{2})^2 + 15^2 = 50 + 225 = 275$$

and looking at the right hand side, we find that

$$(7 \sqrt{5})^2 = 245,$$

and the equation is not true. So the shaded triangle is not a right triangle.

Solution: II. Simpler version of solution I.

In solution I, it is not necessary to take square roots to find the hypotenuses, since it is only the squares of the hypotenuses that are need to verify that the shaded triangle is not a right triangle.

Using the Pythagorean Theorem, we can find the square of the hypotenuse of the smallest right triangle. $$5^2 + 5^2 = 50.$$ Similarly, the square of the hypotenuse of the right triangle with side lengths 7 and 14 is $$7^2+14^2 = 245.$$ The square of the third side of the shaded triangle is $$15^2 = 225.$$

If the shaded triangle is a right triangle, then squares of the side lengths must satisfy Pythagoras's theorem. We have found that those squares are 50, 245, and 225. The largest of these is 245, so the other two must add up to 245. But $50 + 225 = 275 \neq 245$. So the shaded triangle is not a right triangle.

#### Chris Newton says:

almost 5 years

In an effort to begin performance tasks, our school modified this task. Since it didn't specify to use the Pythagorean Theorem, a 5th grader answered this way: since the triangle with sides 5 and 5 has 45 degree angles, then if either of the grey angles adjacent to them measured 90 degrees, the remaining angle adjacent would also need to measure 45 degrees. If they did, the bottom-left and top-right triangles would also be isosceles, but they are not. (Side lengths given.) Therefore the grey angles are not 90 degrees. For the third grey angle (top-left corner), he reasoned that if it measured 90 degrees, it would take up the whole corner.

#### Cam says:

almost 5 years

A remarkable solution! Indeed, it calls into question the use of the Pythagorean Theorem at all! Thanks very much for sharing.

#### Kate says:

Potential alternate solution: I realize that slopes of perpendicular lines isn't an 8th grade standard, but a student might notice the slope of the length sqrt(50) segment is 1, and the slope of the length 15 segment can't be -1, so they can't be perpendicular.