Calculator 6.11 Unit Test: The Pythagorean Theorem – Part 1 Determine whether each set of side lengths could be the sides of a right triangle. Drag and drop each set of side lengths to the correct box. Right Triangle 8 in., 15 in., 17 in. 4 in., 15 in., 17 in. Not a Right Triangle 1 2 3 4 5 6 7 8 9
The Correct Answer and Explanation is:
Here is the correct answer and explanation for the problem.
Correct Answer
- Right Triangle: 8 in., 15 in., 17 in.
- Not a Right Triangle: 4 in., 15 in., 17 in.
Explanation
To determine if a set of side lengths can form a right triangle, we use the converse of the Pythagorean theorem. This theorem states that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. The formula is expressed as a² + b² = c², where ‘a’ and ‘b’ are the lengths of the two shorter sides (the legs), and ‘c’ is the length of the longest side (the hypotenuse).
Let’s test the first set of side lengths: 8 in., 15 in., 17 in.
In this set, the longest side is 17 inches, so we will assign it as ‘c’. The other two sides, 8 inches and 15 inches, will be ‘a’ and ‘b’.
- Let a = 8, b = 15, and c = 17.
- Now, we check if a² + b² = c².
- Calculate the sum of the squares of the two shorter sides: 8² + 15² = 64 + 225 = 289.
- Calculate the square of the longest side: 17² = 289.
- Since 289 equals 289, the equation a² + b² = c² is true. Therefore, the side lengths 8 in., 15 in., and 17 in. form a right triangle.
Next, let’s test the second set of side lengths: 4 in., 15 in., 17 in.
Again, the longest side is 17 inches, which will be our ‘c’. The other two sides are 4 inches and 15 inches.
- Let a = 4, b = 15, and c = 17.
- We check if a² + b² = c².
- Calculate the sum of the squares of the two shorter sides: 4² + 15² = 16 + 225 = 241.
- Calculate the square of the longest side: 17² = 289.
- In this case, 241 does not equal 289. The equation a² + b² = c² is false. Consequently, the side lengths 4 in., 15 in., and 17 in. cannot form a right triangle.
