Question

find the missing side of each triangle. write your answers as decimals to the nearest hundredth.

1. 10.9 in 6.5 in
2. 13.5 ft 6.9 ft

find the missing side of each triangle using the pythagorean theorem. leave your answers in simplest radical form (not a decimal!)

1. 8 mi 4 mi
2. 6 in 4 in

use the pythagorean theorem to determine if the triangle is right, acute, or obtuse - dont go picture!

1. 12 mi 15 mi 9 mi
2. 9 in 5 in 13 in

Explanation:

1. For the first right - triangle with sides 10.9 in and 6.5 in:

• # Explanation:
• ## Step1: Identify the Pythagorean Theorem

The Pythagorean Theorem for a right - triangle is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side) and \(a\) and \(b\) are the other two sides. Here, we assume the missing side \(x\) is the hypotenuse.

• ## Step2: Substitute the values into the formula

\(x=\sqrt{10.9^{2}+6.5^{2}}=\sqrt{118.81 + 42.25}=\sqrt{161.06}\approx12.70\) in

• # Answer:

\(x\approx12.70\) in

1. For the second right - triangle with sides 13.5 ft and 6.9 ft:

• # Explanation:
• ## Step1: Use the Pythagorean Theorem

Assuming the missing side \(x\) is the hypotenuse, by the Pythagorean Theorem \(x=\sqrt{13.5^{2}+6.9^{2}}\).

• ## Step2: Calculate the values

\(x=\sqrt{182.25+47.61}=\sqrt{229.86}\approx15.16\) ft

• # Answer:

\(x\approx15.16\) ft

1. For the third right - triangle with sides 4 mi and 8 mi:

• # Explanation:
• ## Step1: Apply the Pythagorean Theorem

If the missing side \(x\) is the hypotenuse, then \(x=\sqrt{4^{2}+8^{2}}\).

• ## Step2: Simplify the expression

\(x=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}\) mi

• # Answer:

\(x = 4\sqrt{5}\) mi

1. For the fourth right - triangle with sides 4 in and 6 in:

• # Explanation:
• ## Step1: Use the Pythagorean Theorem

Assuming the missing side \(x\) is the hypotenuse, \(x=\sqrt{4^{2}+6^{2}}\).

• ## Step2: Simplify the result

\(x=\sqrt{16+36}=\sqrt{52}=2\sqrt{13}\) in

• # Answer:

\(x = 2\sqrt{13}\) in

1. For the fifth triangle with sides 9 mi, 12 mi, and 15 mi:

• # Explanation:
• ## Step1: Check the Pythagorean relationship

Let \(a = 9\), \(b = 12\), and \(c = 15\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\). \(a^{2}+b^{2}=9^{2}+12^{2}=81 + 144=225\), and \(c^{2}=15^{2}=225\).

• ## Step2: Determine the type of triangle

Since \(a^{2}+b^{2}=c^{2}\), the triangle is a right - triangle.

• # Answer:

The triangle is a right - triangle.

1. For the sixth triangle with sides 5 in, 9 in, and 13 in:

• # Explanation:
• ## Step1: Check the Pythagorean relationship

Let \(a = 5\), \(b = 9\), and \(c = 13\). Calculate \(a^{2}+b^{2}=5^{2}+9^{2}=25 + 81 = 106\), and \(c^{2}=13^{2}=169\).

• ## Step2: Determine the type of triangle

Since \(a^{2}+b^{2}

• # Answer:

The triangle is an obtuse - triangle.

Answer:

1. For the first right - triangle with sides 10.9 in and 6.5 in:

• # Explanation:
• ## Step1: Identify the Pythagorean Theorem

The Pythagorean Theorem for a right - triangle is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side) and \(a\) and \(b\) are the other two sides. Here, we assume the missing side \(x\) is the hypotenuse.

• ## Step2: Substitute the values into the formula

\(x=\sqrt{10.9^{2}+6.5^{2}}=\sqrt{118.81 + 42.25}=\sqrt{161.06}\approx12.70\) in

• # Answer:

\(x\approx12.70\) in

1. For the second right - triangle with sides 13.5 ft and 6.9 ft:

• # Explanation:
• ## Step1: Use the Pythagorean Theorem

Assuming the missing side \(x\) is the hypotenuse, by the Pythagorean Theorem \(x=\sqrt{13.5^{2}+6.9^{2}}\).

• ## Step2: Calculate the values

\(x=\sqrt{182.25+47.61}=\sqrt{229.86}\approx15.16\) ft

• # Answer:

\(x\approx15.16\) ft

1. For the third right - triangle with sides 4 mi and 8 mi:

• # Explanation:
• ## Step1: Apply the Pythagorean Theorem

If the missing side \(x\) is the hypotenuse, then \(x=\sqrt{4^{2}+8^{2}}\).

• ## Step2: Simplify the expression

\(x=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}\) mi

• # Answer:

\(x = 4\sqrt{5}\) mi

1. For the fourth right - triangle with sides 4 in and 6 in:

• # Explanation:
• ## Step1: Use the Pythagorean Theorem

Assuming the missing side \(x\) is the hypotenuse, \(x=\sqrt{4^{2}+6^{2}}\).

• ## Step2: Simplify the result

\(x=\sqrt{16+36}=\sqrt{52}=2\sqrt{13}\) in

• # Answer:

\(x = 2\sqrt{13}\) in

1. For the fifth triangle with sides 9 mi, 12 mi, and 15 mi:

• # Explanation:
• ## Step1: Check the Pythagorean relationship

Let \(a = 9\), \(b = 12\), and \(c = 15\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\). \(a^{2}+b^{2}=9^{2}+12^{2}=81 + 144=225\), and \(c^{2}=15^{2}=225\).

• ## Step2: Determine the type of triangle

Since \(a^{2}+b^{2}=c^{2}\), the triangle is a right - triangle.

• # Answer:

The triangle is a right - triangle.

1. For the sixth triangle with sides 5 in, 9 in, and 13 in:

• # Explanation:
• ## Step1: Check the Pythagorean relationship

Let \(a = 5\), \(b = 9\), and \(c = 13\). Calculate \(a^{2}+b^{2}=5^{2}+9^{2}=25 + 81 = 106\), and \(c^{2}=13^{2}=169\).

• ## Step2: Determine the type of triangle

Since \(a^{2}+b^{2}

• # Answer:

The triangle is an obtuse - triangle.