Question

1. is a triangle with sides of length 5 cm, 12 cm, and 13 cm a right triangle?
2. for what value of r would triangle ant be a right triangle?
3. a tree broke 6 feet from the bottom. if the top landed 12 feet from the base, how tall was the tree before it broke?

Explanation:

Step1: Apply Pythagorean theorem for problem 8

For a right - triangle \(ANT\) with legs \(AN = 2\) mi and \(NT=\sqrt{5}\) mi, and hypotenuse \(AT=r\). By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 2\), \(b=\sqrt{5}\), and \(c = r\).

$$r^{2}=2^{2}+(\sqrt{5})^{2}$$

Step2: Simplify the equation

$$r^{2}=4 + 5$$

$$r^{2}=9$$

$$r = 3$$

(since \(r>0\) as it represents a length)

Step3: Check for problem 7

For a triangle with sides \(a = 5\), \(b = 12\), \(c = 13\). Check if \(a^{2}+b^{2}=c^{2}\)

$$5^{2}+12^{2}=25 + 144=169$$

$$13^{2}=169$$

Since \(5^{2}+12^{2}=13^{2}\), it is a right - triangle.

Step4: Solve problem 9

Let the broken part of the tree be the hypotenuse of a right - triangle. The base \(a = 12\) feet and the unbroken part \(b = 6\) feet. By the Pythagorean theorem, the length of the broken part \(l=\sqrt{6^{2}+12^{2}}=\sqrt{36 + 144}=\sqrt{180}=6\sqrt{5}\) feet. The total height of the tree before it broke is \(h=6 + 6\sqrt{5}\approx6+6\times2.24=6 + 13.44 = 19.44\) feet

Answer:

1. \(r = 3\) mi; 7. Yes; 9. Approximately \(19.44\) feet