Question 4
Construct a triangle with sidelengths in the ratio . Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?
- A ratio of specifies the relative proportions of a triangle's sides, forming a right-angled triangle since .
- The actual side lengths are given by for any positive scale factor .
- Congruence vs. Similarity: Congruent triangles must have both the same shape and the exact same size. Triangles with the same side ratio have the same shape (similar), but vary in size for different values of .
Step 1 · Construct Triangle with Side Ratio
Let the side lengths for construction be chosen with , giving sides , , and .
Note that , forming a right-angled triangle.
Steps of Construction:
- Draw a base line segment .
- With center and radius , draw an arc above .
- With center and radius , draw an arc intersecting the previous arc at .
- Join and .
is the required triangle.
Step 2 · Check Congruence for Different Scale Factors
For any positive multiplier , the sides of the triangle are :
- For : sides are
- For : sides are
- For : sides are
All these triangles have identical shape and angle measures (they are similar), but their side lengths and sizes differ.
Since congruent figures must have identical shape and identical size, triangles with different values of are not congruent.
No, all triangles with sidelength ratio are not congruent to each other. They have the same shape (similar) but can have different sizes depending on the scale factor .
- Confusing Congruence with Similarity: Triangles with the same ratio of sides are similar (same shape), but they are only congruent if they have the exact same side lengths.
- Assuming Unique Triangle Size from a Ratio: A ratio like defines infinitely many triangles of varying sizes depending on the chosen multiplier .
More questions in FIO
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