Question

(b) can these be the side lengths of a triangle? 30 cm, 60 cm, 40 cm
yes
no
triangles with these measurements are not necessarily congruent.

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for three side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the sum of the two shorter sides must be greater than the longest side, i.e., \(a + b>c\).

Step2: Identify the sides

Given side lengths: \(30\) cm, \(40\) cm, and \(60\) cm. The longest side \(c = 60\) cm, and the two shorter sides \(a = 30\) cm, \(b = 40\) cm.

Step3: Check the inequality

Calculate the sum of the two shorter sides: \(30 + 40=70\) cm. Now, compare this sum with the longest side: \(70>60\). Also, we can check the other inequalities (though the most critical is the sum of the two smaller ones against the largest): \(30 + 60>40\) ( \(90>40\), true) and \(40 + 60>30\) ( \(100>30\), true). Since all triangle inequalities are satisfied, these can be the side lengths of a triangle.

Answer:

Yes (because \(30 + 40>60\), \(30 + 60>40\), and \(40 + 60>30\), satisfying the triangle inequality theorem)