Question

1. show whether each triangle in the table is a right triangle.

triangle	side lengths (cm)
b	7, 8, 11
c	7, 24, 25
d	16, 30, 34
e	10, 11, 14

Explanation:

To determine if a triangle is a right triangle, we use the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\) (the legs), and \(c\) (the hypotenuse, the longest side), \(a^2 + b^2 = c^2\). We will check each triangle by squaring the lengths of the sides, adding the squares of the two shorter sides, and comparing the result to the square of the longest side.

Triangle A: Side Lengths 9, 12, 15

Step 1: Identify the longest side

The longest side is 15 cm. So, \(c = 15\), and \(a = 9\), \(b = 12\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 9^2 + 12^2 \\ &= 81 + 144 \\ &= 225 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 15^2 = 225
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(9^2 + 12^2 = 15^2\) (225 = 225), Triangle A is a right triangle.

Triangle B: Side Lengths 7, 8, 11

Step 1: Identify the longest side

The longest side is 11 cm. So, \(c = 11\), and \(a = 7\), \(b = 8\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 7^2 + 8^2 \\ &= 49 + 64 \\ &= 113 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 11^2 = 121
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(7^2 + 8^2
eq 11^2\) (113 ≠ 121), Triangle B is not a right triangle.

Triangle C: Side Lengths 7, 24, 25

Step 1: Identify the longest side

The longest side is 25 cm. So, \(c = 25\), and \(a = 7\), \(b = 24\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 7^2 + 24^2 \\ &= 49 + 576 \\ &= 625 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 25^2 = 625
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(7^2 + 24^2 = 25^2\) (625 = 625), Triangle C is a right triangle.

Triangle D: Side Lengths 16, 30, 34

Step 1: Identify the longest side

The longest side is 34 cm. So, \(c = 34\), and \(a = 16\), \(b = 30\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 16^2 + 30^2 \\ &= 256 + 900 \\ &= 1156 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 34^2 = 1156
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(16^2 + 30^2 = 34^2\) (1156 = 1156), Triangle D is a right triangle.

Triangle E: Side Lengths 10, 11, 14

Step 1: Identify the longest side

The longest side is 14 cm. So, \(c = 14\), and \(a = 10\), \(b = 11\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 10^2 + 11^2 \\ &= 100 + 121 \\ &= 221 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 14^2 = 196
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(10^2 + 11^2
eq 14^2\) (221 ≠ 196), Triangle E is not a right triangle.

Summary:

• Triangle A: Right triangle ( \(9^2 + 12^2 = 15^2\) )
• Triangle B: Not a right triangle ( \(7^2 + 8^2

eq 11^2\) )

• Triangle C: Right triangle ( \(7^2 + 24^2 = 25^2\) )
• Triangle D: Right triangle ( \(16^2 + 30^2 = 34^2\) )
• Triangle E: Not a right triangle ( \(10^2 + 11^2

eq 14^2\) )

Answer:

To determine if a triangle is a right triangle, we use the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\) (the legs), and \(c\) (the hypotenuse, the longest side), \(a^2 + b^2 = c^2\). We will check each triangle by squaring the lengths of the sides, adding the squares of the two shorter sides, and comparing the result to the square of the longest side.

Triangle A: Side Lengths 9, 12, 15

Step 1: Identify the longest side

The longest side is 15 cm. So, \(c = 15\), and \(a = 9\), \(b = 12\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 9^2 + 12^2 \\ &= 81 + 144 \\ &= 225 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 15^2 = 225
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(9^2 + 12^2 = 15^2\) (225 = 225), Triangle A is a right triangle.

Triangle B: Side Lengths 7, 8, 11

Step 1: Identify the longest side

The longest side is 11 cm. So, \(c = 11\), and \(a = 7\), \(b = 8\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 7^2 + 8^2 \\ &= 49 + 64 \\ &= 113 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 11^2 = 121
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(7^2 + 8^2
eq 11^2\) (113 ≠ 121), Triangle B is not a right triangle.

Triangle C: Side Lengths 7, 24, 25

Step 1: Identify the longest side

The longest side is 25 cm. So, \(c = 25\), and \(a = 7\), \(b = 24\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 7^2 + 24^2 \\ &= 49 + 576 \\ &= 625 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 25^2 = 625
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(7^2 + 24^2 = 25^2\) (625 = 625), Triangle C is a right triangle.

Triangle D: Side Lengths 16, 30, 34

Step 1: Identify the longest side

The longest side is 34 cm. So, \(c = 34\), and \(a = 16\), \(b = 30\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 16^2 + 30^2 \\ &= 256 + 900 \\ &= 1156 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 34^2 = 1156
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(16^2 + 30^2 = 34^2\) (1156 = 1156), Triangle D is a right triangle.

Triangle E: Side Lengths 10, 11, 14

Step 1: Identify the longest side

The longest side is 14 cm. So, \(c = 14\), and \(a = 10\), \(b = 11\).

Step 2: Calculate \(a^2 + b^2\)

\[

$$\begin{align*} a^2 + b^2 &= 10^2 + 11^2 \\ &= 100 + 121 \\ &= 221 \end{align*}$$

\]

Step 3: Calculate \(c^2\)

\[
c^2 = 14^2 = 196
\]

Step 4: Compare \(a^2 + b^2\) and \(c^2\)

Since \(10^2 + 11^2
eq 14^2\) (221 ≠ 196), Triangle E is not a right triangle.

Summary:

• Triangle A: Right triangle ( \(9^2 + 12^2 = 15^2\) )
• Triangle B: Not a right triangle ( \(7^2 + 8^2

eq 11^2\) )

• Triangle C: Right triangle ( \(7^2 + 24^2 = 25^2\) )
• Triangle D: Right triangle ( \(16^2 + 30^2 = 34^2\) )
• Triangle E: Not a right triangle ( \(10^2 + 11^2

eq 14^2\) )