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Converse of the Pythagorean Theorem

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Derive and Use the Converse of the Pythagorean Theorem

Do you know how to prove that a triangle is a right triangle? Take a look at this dilemma.

A triangle has side lengths of 6.6, 8.8 and 11. Is this triangle a right triangle?

To figure this out, you will need to know how to use the Converse of the Pythagorean Theorem. Pay attention and you will know how to accomplish this task by the end of the Concept.

Guidance

Do you remember the Pythagorean Theorem? Well first, think about a right triangle and its properties.

The legs are the two sides of the triangle that are labeled a and b . The hypotenuse is the longest side of a right triangle and it is labeled c . There is a special relationship between the legs of a right triangle and the hypotenuse of a right triangle.

One of the special characteristics of right triangles is described by the Pythagorean Theorem , thought to have been developed around 500 B.C.E. It states that the squared value of the hypotenuse will equal the sum of the squares of the two legs. In the triangle above, the sum of the squares of the legs is a^2 + b^2 and the square of the hypotenuse is c^2 . So, the Pythagorean theorem is commonly represented as a^2 + b^2 = c^2 where a and b are the legs of the right triangle and c is the hypotenuse.

The Pythagorean Theorem is known as a^2 + b^2 = c^2 .

You may be asking yourself why that is the case. Well, we can think about the Pythagorean Theorem in terms of a square. We know that there is a relationship between a square and a right triangle. We can divide a square with a diagonal and because a square has four right angles, the diagonal will divide the square into two right triangles. Now because a right triangle comes from the square, the sides will also be related to the square. This is where the Pythagorean Theorem comes from.

If you use logic when thinking about the Pythagorean Theorem, there are many ways that you might find it useful. Always see how your knowledge might be applied to help you on a tough problem.

Take a look at this one.

Classify the triangle below as acute, right, or obtuse.

This triangle is specifically drawn not to scale. Therefore, you cannot decide whether the triangle is acute, right, or obtuse just by looking at it. Take a moment to analyze the side lengths and see how they are related. Two of the sides (15 and 17) are relatively close in length. The third side (8) is about half the length of the two longer sides.

To see if the triangle might be right, try plugging the values into the Pythagorean theorem to see if it makes it true. The hypotenuse is always the longest side, so 17 should be set equal to c . The other two values can represent a and b .

a^2 + b^2 &= c^2\\8^2+15^2&=17^2\\(8 \times 8) + (15 \times 15) &= (17 \times 17)\\64 + 225 &= 289\\289 &= 289

Since both sides of the equation are equal, the Pythagorean Theorem is true.

Therefore, the triangle described in the problem is a right triangle. We can use this logic to determine whether a triangle is a right triangle or not.

Using this logic is referred to as using the converse of the Pythagorean Theorem. The Pythagorean theorem states that in a right triangle, a^2 + b^2 = c^2 .

The converse of the Pythagorean theorem states that if a^2 + b^2 = c^2 , the triangle is a right triangle.

We can use the converse to prove whether or not a triangle is a right triangle.

Remember, if the Pythagorean Theorem works for the values of the triangle, then the triangle is a right triangle. If not, then the triangle is not a right triangle.

Identify whether the triangle below is a right triangle.

This triangle is specifically drawn not to scale. Therefore, you cannot decide whether the triangle is acute, right, or obtuse just by looking at it. Take a moment to analyze the side lengths and see how they are related. Two of the sides (5 and 8) are relatively close in length. The third side (12) is longer.

To see if the triangle might be right, try plugging the values into the Pythagorean theorem to see if it makes it true. The hypotenuse is always the longest side, so 12 should be set equal to c . The other two values can represent a and b .

a^2 + b^2 &= c^2\\5^2 + 8^2 &= 12^2\\(5 \times 5) + (8 \times 8) &= (12 \times 12)\\25 + 64 &= 144\\89 &\ne 144

At the end of the solution, you can see that the result on the left side was 89, and the result on the right side was 144. Therefore, the sum of the squares of the legs did not equal the square of the hypotenuse. So, the triangle is not a right triangle.

Remember to use the Pythagorean Theorem whenever you want to prove that a triangle is or is not a right triangle.

Example A

Is this triangle a right triangle, a = 7, b = 8, c = 15?

Solution: No.

Example B

Is this triangle a right triangle, a = 9, b = 12, c = 18?

Solution: No.

Example C

Is this triangle a right triangle, a = 15, b = 20, c = 25?

Solution: Yes.

Now let's go back to the dilemma from the beginning of the Concept.

If a triangle has side lengths of 6.6, 8.8 and 11, we can substitute these values into the Pythagorean Theorem and by "working backwards", we can prove that this triangle is or is not a right triangle.

a^2 + b^2 = c^2

6.6^2 + 8.8^2 = 11^2

43.56 + 77.44 = 121

121 = 121

Because both sides of this equation are equal, the triangle is a right triangle.

Vocabulary

Right Triangle
one angle is equal to 90^{\circ} .
Legs
the two shorter sides of a right triangle.
Hypotenuse
the longest side of a right triangle.
Pythagorean Theorem
a^2 + b^2 = c^2
Pythagorean Triple
values that work perfectly in the Pythagorean Theorem. The ratio always simplifies to 3:4:5.
Converse of the Pythagorean Theorem
If a^2 + b^2 = c^2 , then the triangle is a right triangle.

Guided Practice

Here is one for you to try on your own.

Is this triangle a right triangle if it has side lengths of 7.5, 10, and 12.5?

Solution

To figure this out, we can use the converse of the Pythagorean Theorem. It states that if the side lengths form a true statement when substituted into the Pythagorean Theorem, then the triangle is a right triangle.

To figure this out, we substitute these side lengths into the Pythagorean Theorem. If the value on the left side of the equation is equal to the value on the right side of the equation, then our triangle is a right triangle.

a^2 + b^2 &= c^2 \\7.5^2 + 10^2 &= 12.5^2 \\56.25 + 100 &= 156.25 \\156.25 &= 156.25

Our triangle is a right triangle.

Video Review

The Pythagorean Theorem and its Converse

Practice

Directions: Think about what you have learned about the Pythagorean Theorem and answer true or false for the following questions.

  1. The Pythagorean Theorem will work for an acute triangle with all 60^{\circ} angles.
  2. The Pythagorean Theorem will work for a right triangle.
  3. The Pythagorean Theorem will only work if the triangle is a right triangle.
  4. The legs of a right triangle are considered the two shorter sides of the right triangle.
  5. The hypotenuse is the longest side of a right triangle.
  6. The converse of the Pythagorean Theorem is used to find the angle measures of an obtuse triangle.
  7. A Pythagorean Triple is when you multiply all of the angle measures by three.
  8. You can use the Pythagorean Theorem to figure out if the side lengths of a triangle make it a right triangle or not.

Directions: Identify whether or not each of the following values is a Pythagorean Triple. Write yes or no for your answer.

  1. 4, 5, 6
  2. 6, 8, 10
  3. 5, 6, 9
  4. 9, 12, 15
  5. 30, 40, 55
  6. 21, 28, 35
  7. 12, 16, 20

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