Question

florian and michelle are trying to construct a triangle. florian says the side lengths could be 6.8, and 10, and the angles of the triangle should have measures of 37°, 53°, and 90°. michelle says the side lengths could be 7, 7, and 14, and the angles should have measures of 35°, 65°, and 80°. which of the friends has the correct measurement for the triangle? how do you know? florian has the correct side lengths because they are all even numbers. florian has the correct side lengths because the measurements fit the triangle inequality rule. michelle has the correct side lengths because they are not all even numbers. michelle has the correct side lengths because their measurements fit the triangle inequality rule.

Explanation:

Step1: Check triangle side - length rule

The triangle inequality rule states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\).
For Florian's side lengths \(a = 6\), \(b = 8\), \(c = 10\):

• \(6+8 = 14>10\)
• \(6 + 10=16>8\)
• \(8+10 = 18>6\)

For Michelle's side lengths \(a = 7\), \(b = 7\), \(c = 14\):

• \(7+7=14\) (violates the triangle inequality rule since \(a + b\) must be greater than \(c\), not equal)

Step2: Check triangle angle - sum rule

The sum of the interior angles of a triangle is \(180^{\circ}\).
For Florian's angles \(37^{\circ}+53^{\circ}+90^{\circ}=187^{\circ}
eq180^{\circ}\)
For Michelle's angles \(35^{\circ}+65^{\circ}+80^{\circ}=180^{\circ}\)

Answer:

Florian has the correct side lengths because the measurements fit the triangle inequality rule.