Answer :
To solve this problem, we need to analyze the given triangle with side lengths of 5 inches, 12 inches, and 13 inches. Let's explore each option carefully:
### Step 1: Check if the triangle is a right triangle
A triangle is a right triangle if the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. This is based on the Pythagorean theorem:
[tex]\[ \text{If } a, b, \text{ and } c \text{ are sides of a triangle and } c \text{ is the hypotenuse:} \][/tex]
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Here, the side lengths are [tex]\(a = 5 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = 13 \)[/tex]:
Calculating the squares:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 13^2 = 169 \][/tex]
Checking the Pythagorean theorem:
[tex]\[ a^2 + b^2 = 25 + 144 = 169 \][/tex]
[tex]\[ c^2 = 169 \][/tex]
Since [tex]\(a^2 + b^2 = c^2\)[/tex], the triangle is a right triangle. Therefore, option C is correct.
### Step 2: Check the other options for completeness
#### Option A: The area of this triangle is 60 in²
To find the area of a right triangle, we can use the formula for the area of a right triangle:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Using [tex]\( a = 5 \)[/tex] and [tex]\( b = 12 \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times 5 \times 12 = \frac{1}{2} \times 60 = 30 \text{ in}^2 \][/tex]
Therefore, option A is incorrect.
#### Option B: It is an acute triangle
An acute triangle has all interior angles less than 90 degrees. Since our triangle is a right triangle (one of its angles is exactly 90 degrees), it cannot be acute. Therefore, option B is incorrect.
#### Option D: The perimeter of this triangle is 25 inches
The perimeter is the sum of all the sides:
[tex]\[ \text{Perimeter} = a + b + c = 5 + 12 + 13 = 30 \text{ inches} \][/tex]
Therefore, option D is incorrect.
### Conclusion
After evaluating all the options, we find that the correct statement is:
C. It is a right triangle.
### Step 1: Check if the triangle is a right triangle
A triangle is a right triangle if the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. This is based on the Pythagorean theorem:
[tex]\[ \text{If } a, b, \text{ and } c \text{ are sides of a triangle and } c \text{ is the hypotenuse:} \][/tex]
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Here, the side lengths are [tex]\(a = 5 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = 13 \)[/tex]:
Calculating the squares:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 13^2 = 169 \][/tex]
Checking the Pythagorean theorem:
[tex]\[ a^2 + b^2 = 25 + 144 = 169 \][/tex]
[tex]\[ c^2 = 169 \][/tex]
Since [tex]\(a^2 + b^2 = c^2\)[/tex], the triangle is a right triangle. Therefore, option C is correct.
### Step 2: Check the other options for completeness
#### Option A: The area of this triangle is 60 in²
To find the area of a right triangle, we can use the formula for the area of a right triangle:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Using [tex]\( a = 5 \)[/tex] and [tex]\( b = 12 \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times 5 \times 12 = \frac{1}{2} \times 60 = 30 \text{ in}^2 \][/tex]
Therefore, option A is incorrect.
#### Option B: It is an acute triangle
An acute triangle has all interior angles less than 90 degrees. Since our triangle is a right triangle (one of its angles is exactly 90 degrees), it cannot be acute. Therefore, option B is incorrect.
#### Option D: The perimeter of this triangle is 25 inches
The perimeter is the sum of all the sides:
[tex]\[ \text{Perimeter} = a + b + c = 5 + 12 + 13 = 30 \text{ inches} \][/tex]
Therefore, option D is incorrect.
### Conclusion
After evaluating all the options, we find that the correct statement is:
C. It is a right triangle.
