https://r2.erweima.ai/mathbot/image/1725457130364mx4xux6f.jpg

The problem asks for the number of scalene triangles with integer side lengths and a perimeter less than 13.

### Scalene Triangle Properties:
- All three sides are of different lengths.
- The sum of the lengths of any two sides must be greater than the length of the third side (Triangle Inequality).

### Possible Perimeters:
Let's consider possible perimeters of scalene triangles where the sum of the sides \( a + b + c < 13 \). The side lengths \( a \), \( b \), and \( c \) must be integers, and the perimeter \( P \) will be the sum of these sides.

#### Perimeter \( P = 12 \):
- \( a + b + c = 12 \)
- Possible integer combinations that are scalene and satisfy the triangle inequality:
  - \( (3, 4, 5) \) → 3 + 4 + 5 = 12

#### Perimeter \( P = 11 \):
- \( a + b + c = 11 \)
- Possible integer combinations that are scalene and satisfy the triangle inequality:
  - \( (2, 4, 5) \) → 2 + 4 + 5 = 11

#### Perimeter \( P = 10 \):
- \( a + b + c = 10 \)
- Possible integer combinations that are scalene and satisfy the triangle inequality:
  - \( (2, 3, 5) \) → 2 + 3 + 5 = 10

#### Perimeter \( P = 9 \):
- \( a + b + c = 9 \)
- Possible integer combinations that are scalene and satisfy the triangle inequality:
  - \( (2, 3, 4) \) → 2 + 3 + 4 = 9

### Summary:
The number of valid scalene triangles with a perimeter less than 13 is four:

1. \( (3, 4, 5) \) → Perimeter = 12
2. \( (2, 4, 5) \) → Perimeter = 11
3. \( (2, 3, 5) \) → Perimeter = 10
4. \( (2, 3, 4) \) → Perimeter = 9

### Answer:
The correct option is **D) 4**.

Would you like to delve into more details or have any questions?

Here are five related questions for further practice:

1. How many right-angled triangles can be formed with integer sides and a perimeter of 30?
2. What is the maximum perimeter of a scalene triangle with integer sides and a side length no greater than 10?
3. If the perimeter of a triangle is 15 and the side lengths are integers, how many scalene triangles can be formed?
4. How many isosceles triangles with integer sides and a perimeter less than 20 can be formed?
5. What is the smallest perimeter for a scalene triangle with integer sides?

**Tip:** When checking possible triangles, always remember the triangle inequality rule—it helps in quickly eliminating invalid side combinations.

Indique el número de triángulos escalenos cuyos lados son enteros, y el perímetro de la región triangular menor que 13.

Learn how to determine the number of scalene triangles with integer sides and a perimeter less than 13. This problem involves geometry concepts such as the triangle inequality theorem and calculations for the perimeter of triangles. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation