https://file.aiquickdraw.com/mathbot/file/1727037126983zx1h2goi.jpg

In the given image, we have a right triangle \( \triangle ABC \) with the following information:

- \( AC = 52 \, \text{cm} \)
- \( CD = 45 \, \text{cm} \)
- \( AB = 25 \, \text{cm} \)

We are asked to determine the perimeter of the triangle \( ABC \) and confirm whether it is a right triangle.

### Step 1: Determine the length of \( BC \)
Using the Pythagorean theorem, we can check if triangle \( ABC \) is a right triangle at \( B \). The Pythagorean theorem states:
\[
AC^2 = AB^2 + BC^2
\]
Substituting the values:
\[
52^2 = 25^2 + BC^2
\]
\[
2704 = 625 + BC^2
\]
\[
BC^2 = 2704 - 625 = 2079
\]
\[
BC = \sqrt{2079} \approx 45.6 \, \text{cm}
\]

### Step 2: Calculate the Perimeter
The perimeter of a triangle is the sum of all its sides:
\[
\text{Perimeter} = AB + BC + AC
\]
Substituting the known values:
\[
\text{Perimeter} = 25 \, \text{cm} + 45.6 \, \text{cm} + 52 \, \text{cm} = 122.6 \, \text{cm}
\]

### Step 3: Conclusion on Right Triangle
Since the Pythagorean theorem holds (the calculations are consistent with the theorem), we can confirm that \( \triangle ABC \) is a right triangle.

### Final Answers:
- The perimeter of triangle \( ABC \) is approximately **122.6 cm**.
- Yes, it is a right triangle.

Would you like further clarification on any step or more details?

Here are 5 related questions for further understanding:
1. How does the Pythagorean theorem help verify right triangles?
2. What is the significance of the hypotenuse in a right triangle?
3. How do you find the area of a right triangle, given its base and height?
4. Can you explain why the sum of squares of the legs equals the square of the hypotenuse?
5. What other methods can be used to verify a triangle's classification?

**Tip**: Always check that the triangle's side lengths satisfy the Pythagorean theorem before confirming if it’s a right triangle!

Determine the perimeter of triangle ABC and state whether or not it is a right triangle, given AC = 52 cm, CD = 45 cm, and AB = 25 cm.

Find the perimeter of triangle ABC and verify if it is a right triangle using the Pythagorean theorem. Given sides are AC = 52 cm, CD = 45 cm, and AB = 25 cm. This problem uses core geometric concepts and is suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation