Question

math 2 unit 3 review right triangle trigonometry
pythagorean theorem: solve for the exact value of the missing segment.
1.
2.

1. how far from the base of the house do you need to place a 15 - foot ladder so that it exactly reaches the top of a 12 - foot tall wall?
2. what is the length of the diagonal of a 10 cm by 15 cm rectangle?
3. do the side lengths of 7, 14, and 16 form a right triangle? how do you know?

Explanation:

Step1: Apply Pythagorean theorem for question 1

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Here \(c = 24\), \(b=6\), and we want to find \(a\). So \(a=\sqrt{c^{2}-b^{2}}=\sqrt{24^{2}-6^{2}}=\sqrt{(24 + 6)(24 - 6)}=\sqrt{30\times18}=\sqrt{540}=6\sqrt{15}\)

Step2: Apply Pythagorean theorem for question 2

Here \(a = 9\), \(b = 12\), and we want to find \(c\). So \(c=\sqrt{a^{2}+b^{2}}=\sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15\)

Step3: Apply Pythagorean theorem for question 3

The ladder is the hypotenuse \(c = 15\) and the height of the wall \(b = 12\). We want to find the distance from the base of the house \(a\). So \(a=\sqrt{c^{2}-b^{2}}=\sqrt{15^{2}-12^{2}}=\sqrt{(15 + 12)(15 - 12)}=\sqrt{27\times3}=\sqrt{81}=9\) feet

Step4: Apply Pythagorean theorem for question 4

For a rectangle with sides \(a = 10\) and \(b = 15\), the diagonal is the hypotenuse of a right - triangle. So \(c=\sqrt{a^{2}+b^{2}}=\sqrt{10^{2}+15^{2}}=\sqrt{100+225}=\sqrt{325}=5\sqrt{13}\text{ cm}\)

Step5: Check for question 5

For a right - triangle, \(a^{2}+b^{2}=c^{2}\). Let \(a = 7\), \(b = 14\), \(c = 16\). Then \(a^{2}+b^{2}=7^{2}+14^{2}=49+196 = 245\) and \(c^{2}=16^{2}=256\). Since \(a^{2}+b^{2}
eq c^{2}\), the side lengths 7, 14, and 16 do not form a right - triangle.

Answer:

1. \(6\sqrt{15}\)
2. \(15\)
3. \(9\) feet
4. \(5\sqrt{13}\text{ cm}\)
5. No, because \(7^{2}+14^{2}=245

eq16^{2}=256\)