Answer :
To determine whether the sides of lengths 18, 45, and 46 can form a triangle and identify what kind of triangle they form, we need to go through a few steps.
### Step 1: Determine if the sides can form a triangle
We use the triangle inequality theorem, which states that for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
- Check [tex]\( a + b > c \)[/tex]:
[tex]\( 18 + 45 > 46 \)[/tex]
[tex]\( 63 > 46 \)[/tex] (True)
- Check [tex]\( a + c > b \)[/tex]:
[tex]\( 18 + 46 > 45 \)[/tex]
[tex]\( 64 > 45 \)[/tex] (True)
- Check [tex]\( b + c > a \)[/tex]:
[tex]\( 45 + 46 > 18 \)[/tex]
[tex]\( 91 > 18 \)[/tex] (True)
Since all three conditions are satisfied, these sides can indeed form a triangle.
### Step 2: Determine the type of triangle
To find out whether the triangle is acute, right, or obtuse, we compare the squares of the sides. Let's denote the sides as [tex]\( a = 18 \)[/tex], [tex]\( b = 45 \)[/tex], and [tex]\( c = 46 \)[/tex]. Without loss of generality, let's assume [tex]\( c \)[/tex] to be the largest side.
1. Square the sides:
[tex]\( a^2 = 18^2 = 324 \)[/tex]
[tex]\( b^2 = 45^2 = 2025 \)[/tex]
[tex]\( c^2 = 46^2 = 2116 \)[/tex]
2. Apply the Pythagorean theorem to classify the triangle:
- Right Triangle: If [tex]\( a^2 + b^2 = c^2 \)[/tex]
- [tex]\( 324 + 2025 = 2349 \)[/tex]
- [tex]\( 2349 \neq 2116 \)[/tex]
- Therefore, it's not a right triangle.
- Acute Triangle: If [tex]\( a^2 + b^2 > c^2 \)[/tex]
- [tex]\( 324 + 2025 = 2349 \)[/tex]
- [tex]\( 2349 > 2116 \)[/tex]
- Therefore, it is an acute triangle.
- Obtuse Triangle: If [tex]\( a^2 + b^2 < c^2 \)[/tex]
- This is not the case here as [tex]\( 2349 > 2116 \)[/tex].
Since [tex]\( a^2 + b^2 > c^2 \)[/tex], the triangle with sides 18, 45, and 46 is an acute triangle.
### Conclusion
The sides of a triangle can have lengths of 18, 45, and 46. The triangle is an acute triangle.
### Step 1: Determine if the sides can form a triangle
We use the triangle inequality theorem, which states that for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
- Check [tex]\( a + b > c \)[/tex]:
[tex]\( 18 + 45 > 46 \)[/tex]
[tex]\( 63 > 46 \)[/tex] (True)
- Check [tex]\( a + c > b \)[/tex]:
[tex]\( 18 + 46 > 45 \)[/tex]
[tex]\( 64 > 45 \)[/tex] (True)
- Check [tex]\( b + c > a \)[/tex]:
[tex]\( 45 + 46 > 18 \)[/tex]
[tex]\( 91 > 18 \)[/tex] (True)
Since all three conditions are satisfied, these sides can indeed form a triangle.
### Step 2: Determine the type of triangle
To find out whether the triangle is acute, right, or obtuse, we compare the squares of the sides. Let's denote the sides as [tex]\( a = 18 \)[/tex], [tex]\( b = 45 \)[/tex], and [tex]\( c = 46 \)[/tex]. Without loss of generality, let's assume [tex]\( c \)[/tex] to be the largest side.
1. Square the sides:
[tex]\( a^2 = 18^2 = 324 \)[/tex]
[tex]\( b^2 = 45^2 = 2025 \)[/tex]
[tex]\( c^2 = 46^2 = 2116 \)[/tex]
2. Apply the Pythagorean theorem to classify the triangle:
- Right Triangle: If [tex]\( a^2 + b^2 = c^2 \)[/tex]
- [tex]\( 324 + 2025 = 2349 \)[/tex]
- [tex]\( 2349 \neq 2116 \)[/tex]
- Therefore, it's not a right triangle.
- Acute Triangle: If [tex]\( a^2 + b^2 > c^2 \)[/tex]
- [tex]\( 324 + 2025 = 2349 \)[/tex]
- [tex]\( 2349 > 2116 \)[/tex]
- Therefore, it is an acute triangle.
- Obtuse Triangle: If [tex]\( a^2 + b^2 < c^2 \)[/tex]
- This is not the case here as [tex]\( 2349 > 2116 \)[/tex].
Since [tex]\( a^2 + b^2 > c^2 \)[/tex], the triangle with sides 18, 45, and 46 is an acute triangle.
### Conclusion
The sides of a triangle can have lengths of 18, 45, and 46. The triangle is an acute triangle.
