https://file.aiquickdraw.com/mathbot/file/172661557284560da29yl.jpg

The problem presented in the image is about a triangle with a perimeter of 90 centimeters, with sides labeled \( \ell \), \( m \), and \( n \). The task is divided into two parts:

### Part a:
You are asked to write and solve a system of linear equations to find the lengths of the sides \( \ell \), \( m \), and \( n \). The relations given in the diagram are:

- \( m = 5n - 5 \)
- \( \ell = 4m - n \)

Also, the perimeter of the triangle is 90 centimeters, so:
\[
\ell + m + n = 90
\]

### Part b:
You are asked to determine if the triangle is a right triangle.

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#### Let's first tackle **part a**:

We have the following system of equations:

1. \( \ell + m + n = 90 \)
2. \( m = 5n - 5 \)
3. \( \ell = 4m - n \)

Now, let's solve this system of equations for \( \ell \), \( m \), and \( n \).

The solution for the lengths of the sides \( \ell \), \( m \), and \( n \) is:

- \( \ell = \frac{337}{5} = 67.4 \) cm
- \( m = 18 \) cm
- \( n = \frac{23}{5} = 4.6 \) cm

### For **part b**:
To determine if the triangle is a right triangle, we can apply the Pythagorean theorem. In a right triangle, the square of the length of the hypotenuse (the longest side) should equal the sum of the squares of the other two sides.

Let's check if this holds true. We'll calculate:

\[
\ell^2 = m^2 + n^2
\]

We'll compute this next.The calculations show:

\[
\ell^2 = \frac{113569}{25} = 4542.76
\]
\[
m^2 + n^2 = \frac{8629}{25} = 345.16
\]

Since \( \ell^2 \neq m^2 + n^2 \), the triangle does not satisfy the Pythagorean theorem and is **not a right triangle**.

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Would you like further details or have any questions?

### Related Questions:
1. How do you solve a system of linear equations using substitution?
2. What is the geometric significance of the Pythagorean theorem in triangles?
3. How can you classify triangles based on their sides and angles?
4. How do you determine the perimeter of various geometric shapes?
5. Can the relations between triangle sides be used in other types of triangles (e.g., isosceles, equilateral)?

### Tip:
Always verify the conditions of geometric properties like the Pythagorean theorem when classifying triangles!

A triangle has a perimeter of 90 centimeters. Use a linear system to determine the lengths of sides l, m, and n. Is the triangle a right triangle?

Learn how to determine the lengths of sides of a triangle with a perimeter of 90 centimeters using a system of linear equations. Find out if the triangle is a right triangle by applying the Pythagorean theorem.

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