Question

which real - world scenario involves a right triangle?
a triangular plot of land with side lengths of 10 yards, 10 yards, and 15 yards
a triangular bathroom tile with side lengths of 6 inches, 8 inches, and 12 inches
a triangular street sign with side lengths of 3 feet, 3 feet, and 3 feet
a triangular bike path with lengths of 5 miles, 12 miles, and 13 miles

Explanation:

Step1: Apply the Pythagorean theorem

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

For the triangular plot of land:

Let \(a = 10\), \(b=10\), \(c = 15\). Then \(a^{2}+b^{2}=10^{2}+10^{2}=100 + 100=200\), and \(c^{2}=15^{2}=225\). Since \(200
eq225\), it is not a right - triangle.

For the bathroom tile:

Let \(a = 6\), \(b = 8\), \(c = 12\). Then \(a^{2}+b^{2}=6^{2}+8^{2}=36+64 = 100\), and \(c^{2}=12^{2}=144\). Since \(100
eq144\), it is not a right - triangle.

For the triangular street sign:

Let \(a = 3\), \(b = 3\), \(c = 3\). Then \(a^{2}+b^{2}=3^{2}+3^{2}=9 + 9=18\), and \(c^{2}=3^{2}=9\). Since \(18
eq9\), it is not a right - triangle.

For the triangular bike path:

Let \(a = 5\), \(b = 12\), \(c = 13\). Then \(a^{2}+b^{2}=5^{2}+12^{2}=25+144 = 169\), and \(c^{2}=13^{2}=169\). Since \(a^{2}+b^{2}=c^{2}\), it is a right - triangle.

Answer:

The triangular bike path with lengths of 5 miles, 12 miles, and 13 miles involves a right - triangle.