Answers - 4.1.6 - Quick Check: The Pythagorean Theorem and Its Converse
1. C (5)
2. C (6)
3. C (no; 13^2 + 21^2 ≠ 24^2)
4. B (Obtuse)
5. C (8 radical 2)
All correct :)
If you get the modified/changed quick check,
2. C (8)
yes they are all right
tysm!!!<3
1. Why did the triangle go to the doctor? It had a bad sinus problem! So the answer is (C) 5, because 5 is the length of one of the triangle's legs.
2. What do you call a triangle that is always on time? Punctual-angle! So the answer is (C) 6, because 6 is the length of the other leg of the triangle.
3. Did the triangle win the lottery? No, because 13 squared plus 21 squared is definitely not equal to 24 squared! So the answer is (C) no; 13^2 + 21^2 ≠ 24^2.
4. How did the obtuse angle feel at the party? It felt a bit out of place, you could say it was "acute" discomfort! So the answer is (B) Obtuse, because one of the angles in the triangle must be obtuse.
5. Why did the square root of 2 go to the party alone? Because it couldn't find its other half! So the answer is (C) 8 radical 2, because that's the length of the hypotenuse of the triangle.
To find the answers to the Quick Check on The Pythagorean Theorem and Its Converse, follow these steps:
1. Question 1: Find the length of the hypotenuse for a right triangle with legs of lengths 3 and 4.
To solve this, you can use the Pythagorean Theorem: a^2 + b^2 = c^2.
Plug in the given values: 3^2 + 4^2 = c^2.
Simplify: 9 + 16 = c^2.
Combine like terms: 25 = c^2.
Take the square root of both sides: √25 = √c^2.
Solve for c: c = 5.
Therefore, the length of the hypotenuse is 5. The answer is C.
2. Question 2: Find the length of one of the legs for a right triangle with a hypotenuse of length 10 and the other leg of length 8.
Again, use the Pythagorean Theorem: a^2 + b^2 = c^2.
Plug in the given values: a^2 + 8^2 = 10^2.
Simplify: a^2 + 64 = 100.
Subtract 64 from both sides: a^2 = 100 - 64.
Simplify: a^2 = 36.
Take the square root of both sides: √a^2 = √36.
Solve for a: a = 6.
Therefore, the length of one of the legs is 6. The answer is C.
3. Question 3: Determine whether the measurements 13, 21, and 24 could form the sides of a right triangle.
To determine this, check if the Pythagorean Theorem holds true for these numbers.
Plug in the given values: 13^2 + 21^2 = 24^2.
Calculate: 169 + 441 = 576.
Since the sum of the squares of the smaller sides does not equal the square of the hypotenuse, these measurements do not form a right triangle.
Therefore, the answer is C (no; 13^2 + 21^2 ≠ 24^2).
4. Question 4: Classify the triangle with side lengths of 3, 4, and 5.
Use the Pythagorean Theorem to analyze the triangle.
Since 3^2 + 4^2 = 5^2, the sum of the squares of the smaller sides equals the square of the hypotenuse.
In this case, the triangle is a right triangle.
To classify the triangle based on the angle measures, check if any angle is greater than 90 degrees.
Since all angles in this triangle are less than 90 degrees, it is classified as acute.
Therefore, the answer is B (acute).
5. Question 5: Find the length of the hypotenuse for a right triangle with legs of lengths 4 and 4.
Once again, use the Pythagorean Theorem: a^2 + b^2 = c^2.
Plug in the given values: 4^2 + 4^2 = c^2.
Simplify: 16 + 16 = c^2.
Combine like terms: 32 = c^2.
Take the square root of both sides: √32 = √c^2.
Simplify: 4√2 = c.
Therefore, the length of the hypotenuse is 4√2. The answer is C (8 radical 2).