Questions: Determine if a triangle can be formed with the given lengths. If so, classify the triangle by angles.

Transcript text: Determine if a triangle can be formed with the given lengths. If so, classify the triangle by angles.

Solution

Solution Steps

To determine if a triangle can be formed with the given lengths, we need to check if the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality Theorem). If a triangle can be formed, we then use the Converse of the Pythagorean Theorem to classify the triangle by its angles: if \(a^2 + b^2 = c^2\), it's a right triangle; if \(a^2 + b^2 > c^2\), it's an acute triangle; and if \(a^2 + b^2 < c^2\), it's an obtuse triangle.

Step 1: Determine Triangle Type for \( (3, 4, 5) \)

For the sides \( a = 3 \), \( b = 4 \), and \( c = 5 \):

Check if a triangle can be formed: \( 3 + 4 > 5 \) (True).
Classify the triangle: \[ a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 \] \[ c^2 = 5^2 = 25 \] Since \( a^2 + b^2 = c^2 \), the triangle is a right triangle.

Step 2: Determine Triangle Type for \( (2, 2, 3) \)

For the sides \( a = 2 \), \( b = 2 \), and \( c = 3 \):

Check if a triangle can be formed: \( 2 + 2 > 3 \) (True).
Classify the triangle: \[ a^2 + b^2 = 2^2 + 2^2 = 4 + 4 = 8 \] \[ c^2 = 3^2 = 9 \] Since \( a^2 + b^2 < c^2 \), the triangle is an obtuse triangle.

Step 3: Determine Triangle Type for \( (3, 3, 5) \)

For the sides \( a = 3 \), \( b = 3 \), and \( c = 5 \):

Check if a triangle can be formed: \( 3 + 3 > 5 \) (True).
Classify the triangle: \[ a^2 + b^2 = 3^2 + 3^2 = 9 + 9 = 18 \] \[ c^2 = 5^2 = 25 \] Since \( a^2 + b^2 < c^2 \), the triangle is also an obtuse triangle.