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 =