The Pythagorean Theorem

The Pythagorean Theorem exercise

• Define the Pythagorean Theorem.

  Hints

  The Pythagorean theorem focuses on the side lengths of right triangles.

  The Pythagorean Theorem focuses on the relationships between the side lengths of right triangles.

  Some options in the answer set are true statements about triangles, but are not part of the Pythagorean Theorem.

  Solution

  The Pythagorean Theorem relates the side lengths of right triangles. The theorem states that the side lengths of right triangles have a special relationship with each other that is described by the equation $a^2 + b^2 = c^2$. Further, there are sets of integers that satisfy this equation, which are known as "Pythagorean Triples".

  Knowing this, let's evaluate each of the statements and determine if they describe the Pythagorean theorem.

  • The three side lengths of right triangles form sets called "Pythagorean Triples." ✓ This is correct, and is part of the Pythagorean Theorem.

  • A right triangle must have two acute angles. ✗ Although this statement is true, it is not part of the Pythagorean Theorem. This theorem focuses on the side lengths of a right triangle, not the other angles in a right triangle.

  • If a triangle is a right triangle, the side lengths $a$, $b$, and $c$, will satisfy the equation $a^2 + b^2 = c^2$. ✓ This is correct, and is part of the Pythagorean Theorem.

  • The hypotenuse will always be the longest side of a right triangle. ✗ Although this statement is true, it is not part of the Pythagorean Theorem. It can be proved using the Pythagorean Theorem, though!

• Determine if the side lengths are a Pythagorean Triple.

  Hints

  Remember that the Pythagorean Theorem can be represented as an equation that uses the squares of the side lengths of a triangle.

  The Pythagorean Triples must not include fractions or decimals.

  The set of side lengths is a Pythagorean Triple, if they satisfy the equation $a^2 + b^2 = c^2$. This means that the left hand side of the equation must equal the right hand side.

  Solution

  The equation given by the Pythagorean Theorem is $a^2 + b^2 = c^2$. Integers that satisfy these equations are known as "Pythagorean Triples".

  We are given the side lengths of the triangle: 8, 15, and 17. All being whole numbers, they are definitely also integers. So if the set of numbers satisfies the equation $a^2 + b^2 = c^2$, then it must be a Pythagorean Triple.

  The value $c$ is the length of the hypotenuse, which is the longest side. It is also the side across from the right angle. Therefore this side is 17 feet long. Either of the other two sides can act as $a$ or $b$, since they are interchangeable.

  Let's assign the values $a = 8$, $b = 15$, and $c = 17$.

  Substituting these values in, we get:

  $a^2 + b^2 = c^2$

  $8^2 + 15^2 = 17^2$

  $64 + 225 = 289$

  $289 = 289$

  The left and right hand sides of the equation are equal. This means that the values satisfy the equation. We can see that this triangle does have side lengths that are a Pythagorean Triple.

• Find the areas.

  Hints

  How can you use the Pythagorean Theorem to help you with this question?

  The area of each square that is connected to a side represents the literal "square" of that side length. This is equal to the side length, squared.

  What is the relationship between the square of each side length?

  You can use the equation $a^2 + b^2 = c^2$ to find the area of the square that is attached to the side that has an unknown length.

  Solution

  In each of the two cases, we have a right triangle, and we know the lengths of two sides. We must find the area of the square that is connected to the hypotenuse. The area of each square that is connected to a side represents the literal "square" of that side length. This is equal to the side length, squared. We do not know the length of the hypotenuse, but there is another way to find this area.

  From the Pythagorean Theorem, we know that:

  $a^2 + b^2 = c^2$

  We know that $c^2$ represents the area that we must find. We know the side lengths $a$ and $b$.

  $a^2 = 3^2$

  $a^2 = 9$

  $b^2 = 4^2$

  $b^2 = 16$

  Now we can find $c^2$:

  $a^2 + b^2 = c^2$

  $9 + 16 = c^2$

  $25 = c^2$

  Similarly, for the second triangle...

  $a^2 = 36$

  $b^2 = 64$

  $100 = c^2$

• Decide if the ladder fits.

  Hints

  Remember: the hypotenuse is always the longest side of the a right triangle.

  If the equation is satisfied by the given values, then the ladder is the perfect length to fit between the ground and the roof, with the base of the ladder sitting 10 feet from the cottage.

  In this question we move straight from the equation $a^2 + b^2 = c^2$, to the square of the values we substitute in. For example, if we had $a = 3$, $b = 4$, and $c = 5$, we would write:

  $a^2 + b^2 = c^2$

  $9 + 16 = 25$

  By doing this, we are skipping the middle step:

  $3^2 + 4^2 = 5^2$

  Solution

  The ladder, the cottage, and the ground form a right triangle. We are certain about the length of two of the sides of this triangle. We know that the cottage is 24 feet tall. And we know that the bottom of the ladder must sit at least 10 feet from the base of the cottage. The farther away from the cottage the base of the ladder is placed, the longer the ladder will need to be. Let's assume we place the base of the ladder exactly 10 feet from the cottage, so that Damo is able to use the shortest possible ladder.

  If the ladder is the correct length, then this set of side lengths should satisfy the Pythagorean theorem, $a^2 + b^2 = c^2$. We know that $c$ should be the hypotenuse. We can see that the ladder, which is 26 feet long, is opposite the right angle. Additionally, the ladder forms the longest side of this right triangle. Therefore it must be the hypotenuse. The other two sides can be labelled $a$ and $b$ interchangably.

  $a = 10$

  $b = 24$

  $c = 26$

  By substituting the values in, and simplifying the equation, we can check if the equation is satisfied by these values:

  $\begin{array}{rclcl} a^2 + b^2 & = & c^2 &|&\text{subsituting values} \\ 10^2 + 24^2 & = & 26^2 \\ 100 + 576 & = & 676 \\ 676 & = & 676 &|&\text{✓ the two sides are equal} \\ \end{array}$

  The two sides of this equation are equal. Therefore the values do satisfy the equation, and the ladder is the correct length.

• Identify the right triangles.

  Hints

  How can you determine if a triangle is a right triangle using the Pythagorean Theorem?

  What equation can you use to check if a triangle is right, using its side lengths, $a$, $b$, and $c$?

  If a triangle's side lengths, $a$, $b$, and $c$, satisfy the equation $a^2 + b^2 = c^2$, then it is a right triangle.

  Solution

  We know that if a triangle is right, it's side lengths, $a$, $b$, and $c$, must satisfy the equation:

  $a^2 + b^2 = c^2$

  We are given the side lengths for each triangle. Let's use the equation to check if the triangles are right angle triangles.

  The side lengths of the first triangle are 6, 8, and 10. We know that the value for $c$ must be the longest side of the triangle.

  So let's assign the values $a = 6$, $b = 8$, and $c = 10$.

  Putting these values into the equation gives us:

  $6^2 + 8^2 = 10^2$

  $36 + 64 = 100$

  $100 = 100$

  Since the right hand side is equal to the left hand side, the values of $a$, $b$, and $c$ for this triangle satisfy the equation. That means that this is a right triangle!

  Similarly, for the other triangles, we arrive at:

  $a = 9$, $b = 40$, $c = 41$:

  $81 + 1600 = 1681$ ✓ The left and right sides of this equation are equal, so the triangle is right.

  $a = 25$, $b = 60$, $c = 65$:

  $625 + 3600 = 4225$ ✓ The left and right sides of this equation are equal, so the triangle is right.

  $a = 11$, $b = 60$, $c = 61$:

  $121 + 3600 = 3721$ ✓ The left and right sides of this equation are equal, so the triangle is right.

  $a = 26$, $b = 36$, $c = 45$:

  $676 + 1296 = 2025$ ✗ The left and right sides of this equation are not equal, so the triangle is not right.

  $a = 40$, $b = 75$, $c = 85$:

  $1600 + 5625 = 7225$ ✓ The left and right sides of this equation are equal, so the triangle is right.

• Identify the Pythagorean Triples.

  Hints

  Sometimes, the two sides of the simplified equation will not be equal. When that happens, the set is not a Pythagorean triple.

  Be sure to carefully check that you have squared the terms correctly.

  Solution

  Damo wants to know if these sets of numbers are Pythagorean Triples. We can check this using the equation $a^2 + b^2 = c^2$, where $c$ is the largest number in the set. $a$ and $b$ are the other two numbers, and are interchangeable.

  Looking at the first set, $5, 4$ and $6$, let's set:

  $a = 5$

  $b = 4$

  $c = 6$ (because $6$ is the largest number)

  We can substitute these values into the equation:

  \begin{array}{rclcl} a^2 + b^2 & = & c^2 &|&\text{substituting} \\ 5^2 + 4^2 & = & 6^2 &|&\text{simplifying} \\ 25 + 16 & = & 36 &|&\text{simplifying} \\ 41 & = & 36 &|&\text{✗ the two sides of the equation are not equal} \\ \end{array}

  Therefore this set of numbers does not satisfy the equation, and these numbers are not a Pythagorean Triple. So the statement in the question is false.

  Similarly, for the second set, $12, 16$ and $20$:

  $12^2 + 16^2 = 20^2$

  $144 + 256 = 400$

  Both sides of this equation are equal. Therefore the set is a Pythagorean Triple.

  And for the third set, $15, 20$ and $25$:

  $15^2 + 20^2 = 25^2$

  $225 + 400 = 625$

  Both sides of this equation are equal. Therefore the set is a Pythagorean Triple.