Question

determine whether each triangle with sides of given lengths is a right triangle. justify your answer. (example 3)

1. 28 yd, 195 yd, 197 yd

5 30 cm, 122 cm, 125 cm

1. calculate the length of the diagonal of the state of wyoming.

write an equation you could use to find the length of the missing side of each right triangle. then find the missing length. round to the nearest tenth if necessary.
7 a = 48 yd; b = 55 yd

1. a = 23 in.; b = 18 in.
2. b = 5.1 m; c = 12.3 m

Explanation:

Step1: Recall Pythagorean theorem

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

Step2: Check triangle with sides 28 yd, 195 yd, 197 yd

Calculate \(28^{2}+195^{2}=784 + 38025=38809\), and \(197^{2}=38809\). Since \(28^{2}+195^{2}=197^{2}\), it is a right - triangle.

Step3: Check triangle with sides 30 cm, 122 cm, 125 cm

Calculate \(30^{2}+122^{2}=900+14884 = 15784\), and \(125^{2}=15625\). Since \(30^{2}+122^{2}
eq125^{2}\), it is not a right - triangle.

Step4: Calculate diagonal of Wyoming

Using Pythagorean theorem for a rectangle with sides 365 mi and 275 mi. Let \(a = 275\) and \(b = 365\), then \(d^{2}=275^{2}+365^{2}\), \(d=\sqrt{275^{2}+365^{2}}=\sqrt{75625 + 133225}=\sqrt{208850}\approx457.0\) mi.

Step5: For right - triangle with \(a = 48\) yd and \(b = 55\) yd

The equation is \(c^{2}=a^{2}+b^{2}\), \(c=\sqrt{48^{2}+55^{2}}=\sqrt{2304 + 3025}=\sqrt{5329}=73\) yd.

Step6: For right - triangle with \(a = 23\) in and \(b = 18\) in

The equation is \(c^{2}=a^{2}+b^{2}\), \(c=\sqrt{23^{2}+18^{2}}=\sqrt{529+324}=\sqrt{853}\approx29.2\) in.

Step7: For right - triangle with \(b = 5.1\) m and \(c = 12.3\) m

The equation is \(a^{2}=c^{2}-b^{2}\), \(a=\sqrt{c^{2}-b^{2}}=\sqrt{12.3^{2}-5.1^{2}}=\sqrt{(12.3 + 5.1)(12.3 - 5.1)}=\sqrt{17.4\times7.2}=\sqrt{125.28}\approx11.2\) m.

Answer:

• The triangle with sides 28 yd, 195 yd, 197 yd is a right - triangle.
• The triangle with sides 30 cm, 122 cm, 125 cm is not a right - triangle.
• The length of the diagonal of Wyoming is approximately 457.0 mi.
• For \(a = 48\) yd and \(b = 55\) yd, the missing side \(c = 73\) yd.
• For \(a = 23\) in and \(b = 18\) in, the missing side \(c\approx29.2\) in.
• For \(b = 5.1\) m and \(c = 12.3\) m, the missing side \(a\approx11.2\) m.