UpStudy Free Solution:

To find the length of \(CD\) in the given diagram, we can use the Pythagorean theorem. The diagram shows a right triangle with \(CB\) as the base, \(BD\) as the height, and \(CD\) as the hypotenuse.

Given:

\[CB = 8 \space \text { cm} \]

\[BD = 6 \space \text { cm} \]

The Pythagorean theorem states:

\[CD^ 2 = CB^ 2 + BD^ 2\]

Substitute the given values:

\[CD^ 2 = 8^ 2 + 6^ 2\]

\[CD^ 2 = 64 + 36\]

\[CD^ 2 = 100\]

Taking the square root of both sides:

\[CD = \sqrt { 100} \]

\[CD = 10 \space \text { cm} \]

Therefore, the length of \(CD\) is \(10 \space \text { cm} \).

Supplemental Knowledge

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)).

Key Concepts:

1. Pythagorean Theorem Formula:

\[c^ 2 = a^ 2 + b^ 2\]

- \(c\) is the length of the hypotenuse.

- \(a\) and \(b\) are the lengths of the other two sides.

2. Application Steps:

- Identify which side is the hypotenuse (the side opposite the right angle).

- Square the lengths of both legs.

- Add these squares together.

- Take the square root of this sum to find the length of the hypotenuse.

Additional Example:

Consider another right triangle with legs measuring 9 cm and 12 cm. To find the hypotenuse (\(c\)):

\[c^ 2 = 9^ 2 + 12^ 2\]

\[c^ 2 = 81 + 144\]

\[c^ 2 = 225\]

Taking the square root:

\[c = \sqrt { 225} \]

\[c = 15 \space \text { cm} \]

Thus, for this triangle, \(c\) (the hypotenuse) is \(15 \space \text { cm} \).

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