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To determine whether triangle \( ABC \) is a right triangle, we need to analyze the given sides and apply the Pythagorean theorem if appropriate.

From the diagram:
- \( AB = 25 \, \text{cm} \)
- \( AC = 15 \, \text{cm} \)
- \( CD = 12 \, \text{cm} \)
- \( \angle DCA = 90^\circ \), indicating that \( \triangle ACD \) is a right triangle.

### Step-by-Step Solution

1. **Check Triangle \( ACD \):**
   Since \( \triangle ACD \) is a right triangle, we can use the Pythagorean theorem to verify the length \( AC \):
   \[
   AC^2 = AD^2 + CD^2
   \]
   where \( CD = 12 \, \text{cm} \) and \( AD \) is the side opposite \( \angle D \) in triangle \( ACD \).

2. **Calculate \( AD \) (Using the Pythagorean theorem):**
   Since \( AC \) is already known as \( 15 \, \text{cm} \), we can calculate \( AD \) as follows:
   \[
   AD^2 = AC^2 - CD^2
   \]
   Plugging in the values:
   \[
   AD^2 = 15^2 - 12^2 = 225 - 144 = 81
   \]
   \[
   AD = \sqrt{81} = 9 \, \text{cm}
   \]

3. **Verify Triangle \( ABC \):**
   Now, check if \( \triangle ABC \) is a right triangle by using the Pythagorean theorem:
   \[
   AB^2 = AC^2 + BC^2
   \]
   Given \( AB = 25 \, \text{cm} \) and \( AC = 15 \, \text{cm} \), calculate \( BC \) based on these values.

Since \( BC = 

Is triangle ABC in the diagram below a right triangle? Explain.

Learn how to determine if triangle ABC is a right triangle using given side lengths and the Pythagorean theorem. This problem explores the relationship between sides and angles in geometric figures, suitable for Grades 7-10.

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