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Applying the Pythagorean Theorem in a mathematical context


Alignments to Content Standards: 8.G.B

Task

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.

Image_1_0c8e68bded99ca50033612e2fdda9330

Is the shaded triangle a right triangle? Provide a proof for your answer.

Solutions

Solution: I. Proof by contradiction

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:

over 4 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:

over 4 years

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

Kate says:

almost 5 years

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.

Richard Askey says:

over 5 years

The solution is messier than it has to be. There is no reason to consider square roots since it is the squares of the lengths of the sides of the shaded triangle which will be used to see if it is a right triangle.

Bill says:

over 5 years

Thanks Dick, this is an excellent suggestion. I've added second solution.