https://file.aiquickdraw.com/mathbot/file/1729521268677sjyn2fxg.jpg

The image shows a right triangle with a height of 12 inches, a hypotenuse of 15 inches, and the base labeled as \( b \). The goal is to calculate the perimeter of the triangle, which is the sum of all its sides.

### Step 1: Find the length of \( b \)
We can use the Pythagorean theorem, which states:

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

where \( a \) is the height (12 inches), \( b \) is the unknown base, and \( c \) is the hypotenuse (15 inches).

Substituting the given values:

\[
12^2 + b^2 = 15^2
\]

\[
144 + b^2 = 225
\]

\[
b^2 = 225 - 144
\]

\[
b^2 = 81
\]

\[
b = \sqrt{81} = 9
\]

So, the base \( b \) is 9 inches.

### Step 2: Calculate the perimeter
The perimeter \( P \) of the triangle is the sum of all sides:

\[
P = a + b + c = 12 + 9 + 15 = 36 \text{ inches}
\]

### Final Answer:
The perimeter of the triangle is **36 inches**.

Would you like more details or have any questions?

Here are some related questions for deeper understanding:
1. How do you apply the Pythagorean theorem to find missing sides in other triangles?
2. What is the relationship between the perimeter and area of a right triangle?
3. How would you solve this if the triangle was not a right triangle?
4. Can you explain why the Pythagorean theorem only works for right triangles?
5. How does the concept of a perimeter apply to irregular shapes?

**Tip:** The Pythagorean theorem is a powerful tool in right triangles, so always check for a 90-degree angle when using it.

What is the perimeter of a right triangle with a height of 12 inches, a hypotenuse of 15 inches, and an unknown base?

Learn how to calculate the perimeter of a right triangle with a height of 12 inches, a hypotenuse of 15 inches, and an unknown base. This solution applies the Pythagorean theorem and perimeter formula. Suitable for Grades 7-9.

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