Given side lengths of \( 40 \mathrm{~cm}, 70 \mathrm{~cm} \), and 80 cm , how many triangles can be constructed? none more than one

To determine how many triangles can be constructed with the given side lengths of \( 40 \, \text{cm} \), \( 70 \, \text{cm} \), and \( 80 \, \text{cm} \), we can use the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \): 1. \( a + b > c \) 2. \( a + c > b \) 3. \( b + c > a \) Let's denote the sides as follows: - \( a = 40 \, \text{cm} \) - \( b = 70 \, \text{cm} \) - \( c = 80 \, \text{cm} \) Now, we will check the triangle inequalities: 1. \( a + b > c \): \[ 40 + 70 > 80 \implies 110 > 80 \quad \text{(True)} \] 2. \( a + c > b \): \[ 40 + 80 > 70 \implies 120 > 70 \quad \text{(True)} \] 3. \( b + c > a \): \[ 70 + 80 > 40 \implies 150 > 40 \quad \text{(True)} \] Since all three inequalities are satisfied, it means that a triangle can be constructed with the given side lengths. Now, let's consider if more than one triangle can be formed. Since the side lengths are fixed and satisfy the triangle inequality, only one unique triangle can be formed with these specific lengths. Thus, the answer is: - **none** (for more than one triangle) - **one** (for the unique triangle that can be formed) In conclusion, only one triangle can be constructed with the given side lengths.