Construct a triangle with sidelengths in the ratio 3 : 4 : 5. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?
No, all triangles drawn with sidelengths in the ratio 3:4:5 will not be congruent to each other.
They will be similar but not necessarily congruent.
Example: Triangle 1: sides 3 cm, 4 cm, 5 cm Triangle 2: sides 6 cm, 8 cm, 10 cm Both have the same ratio 3:4:5 but different actual lengths.
Reason: Congruent triangles must have exactly equal corresponding sides, not just proportional sides.
Explanation
The chapter discusses ratios and proportional relationships. Two triangles with the same ratio of sidelengths are similar (same shape) but not necessarily congruent (same size). Congruence requires identical side lengths, while similarity only requires proportional sides. The 3:4:5 ratio produces right-angled triangles, but their actual sizes can vary.
Solution Steps
No, all triangles drawn with sidelengths in the ratio 3:4:5 will not be congruent to each other.
They will be similar but not necessarily congruent.
Example: Triangle 1 has sides 3 cm, 4 cm, 5 cm; Triangle 2 has sides 6 cm, 8 cm, 10 cm.
Both have the same ratio 3:4:5 but different actual lengths.
Congruent triangles must have exactly equal corresponding sides, not just proportional sides.