We checked by construction that there are no triangles having sidelengths 3 cm, 4 cm and 8 cm; and 2 cm, 3 cm and 6 cm. Check if you could have found this without trying to construct the triangle.
Rule: Triangle Inequality states that for a triangle to exist, the longest side must be less than the sum of the other two sides.
For lengths 3 cm, 4 cm, 8 cm: Longest side = 8 cm Sum of other two = 3 + 4 = 7 cm 8 > 7 ∴ Triangle cannot be formed.
For lengths 2 cm, 3 cm, 6 cm: Longest side = 6 cm Sum of other two = 2 + 3 = 5 cm 6 > 5 ∴ Triangle cannot be formed.
Conclusion: In both cases, the longest side exceeds the sum of the other two sides, violating the triangle inequality.
Explanation
The chapter introduces the triangle inequality: for a triangle to exist, each length must be smaller than the sum of the other two lengths. The chapter explains that we only need to check if the longest side is less than the sum of the other two sides. If the longest side is greater than or equal to the sum of the other two, the arcs from the endpoints of the base will not intersect, and no triangle can be formed.
Solution Steps
Apply the triangle inequality: the longest side must be less than the sum of the other two sides for a triangle to exist.
For lengths 3, 4, 8: Longest side = 8, sum of other two = 3 + 4 = 7
Since 8 > 7, the triangle inequality is violated and no triangle can be formed.
For lengths 2, 3, 6: Longest side = 6, sum of other two = 2 + 3 = 5
Since 6 > 5, the triangle inequality is violated and no triangle can be formed.