Question

14 multiple choice 2 points which of the following is not a pythagorean triple? 2uv, u² - v², u² + v², where u > v 5, 12, 13 2, 3, 4 3k, 4k, 5k, where k > 0 15 multiple choice 2 points which list shows the lengths of three line segments that could not form a triangle? 5 cm, 6 cm, 7 cm 6 cm, 8 cm, 10 cm 3 cm, 6 cm, 6 cm 4 cm, 7 cm, 13 cm 16 multiple choice 2 points jennies math tutor drew three lines to create a right angle scalene triangle in her math notebook. which of the following are the possible whole number lengths of the three segments? 16, 17, 23 7, 24, 25 2, 3, 4 33, 35, 48

Explanation:

Step1: Recall Pythagorean - theorem for Pythagorean triples

For a Pythagorean triple \((a,b,c)\) (where \(c\) is the longest side), \(a^{2}+b^{2}=c^{2}\).

Step2: Check option 1

For \(a = 2uv\), \(b=u^{2}-v^{2}\), \(c = u^{2}+v^{2}\), \(a^{2}+b^{2}=(2uv)^{2}+(u^{2}-v^{2})^{2}=4u^{2}v^{2}+u^{4}-2u^{2}v^{2}+v^{4}=u^{4}+2u^{2}v^{2}+v^{4}=(u^{2}+v^{2})^{2}=c^{2}\), so it is a Pythagorean triple.

Step3: Check option 2

For \(a = 5\), \(b = 12\), \(c = 13\), \(a^{2}+b^{2}=5^{2}+12^{2}=25 + 144=169=13^{2}=c^{2}\), so it is a Pythagorean triple.

Step4: Check option 3

For \(a = 2\), \(b = 3\), \(c = 4\), \(a^{2}+b^{2}=2^{2}+3^{2}=4 + 9 = 13
eq4^{2}=16\), so it is not a Pythagorean triple.

Step5: Check option 4

For \(a = 3k\), \(b = 4k\), \(c = 5k\) (\(k>0\)), \(a^{2}+b^{2}=(3k)^{2}+(4k)^{2}=9k^{2}+16k^{2}=25k^{2}=(5k)^{2}=c^{2}\), so it is a Pythagorean triple.

Step6: Recall triangle - inequality theorem for question 15

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Step7: Check option 1 for question 15

\(5 + 6>7\), \(5+7>6\), \(6 + 7>5\), so they can form a triangle.

Step8: Check option 2 for question 15

\(6+8>10\), \(6 + 10>8\), \(8+10>6\), so they can form a triangle.

Step9: Check option 3 for question 15

\(3+6>6\), \(3 + 6>6\), \(6+6>3\), so they can form a triangle.

Step10: Check option 4 for question 15

\(4+7=11<13\), so they cannot form a triangle.

Step11: Recall Pythagorean - theorem for question 16

We need to check which set of numbers satisfies \(a^{2}+b^{2}=c^{2}\) for a right - triangle.

Step12: Check option 1 for question 16

\(16^{2}+17^{2}=256+289 = 545
eq23^{2}=529\), not a right - triangle.

Step13: Check option 2 for question 16

\(7^{2}+24^{2}=49+576=625=25^{2}\), it is a right - triangle and scalene.

Step14: Check option 3 for question 16

\(2^{2}+3^{2}=4 + 9=13
eq4^{2}=16\), not a right - triangle.

Step15: Check option 4 for question 16

\(33^{2}+35^{2}=1089+1225=2314
eq48^{2}=2304\), not a right - triangle.

Answer:

1. C. 2, 3, 4
2. D. 4 cm, 7 cm, 13 cm
3. B. [7, 24, 25]