Question

mr hirschler has several lengths of steel with which to construct a triangle. he will use steel pieces with length of 30 inches and 45 inches. determine which pieces of steel found in the table below that can be welded together in the form of a triangle. 10 inches 40 inches 80 inches

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\). Let \(a = 30\) and \(b=45\). Then \(a + b=30 + 45=75\), \(b - a=45 - 30 = 15\). So the length \(c\) of the third side must satisfy \(15

Step2: Check each value

• For \(c = 10\):

Since \(10<15\), it does not satisfy \(15b\)).

• For \(c = 40\):

Since \(15<40<75\). Check triangle - inequality: \(30 + 40=70>45\), \(30+45 = 75>40\), \(40 + 45=85>30\).

• For \(c = 80\):

Since \(80>75\), it does not satisfy \(15c\)).

Answer:

• 10 inches: No. Because \(30 + 10=40<45\) (violates triangle - inequality theorem).
• 40 inches: Yes. Because \(30 + 40>45\), \(30 + 45>40\), \(40 + 45>30\).
• 80 inches: No. Because \(30+45 = 75<80\) (violates triangle - inequality theorem).