Question

Are the shapes similar? (1) All pairs of corresponding sides are in the same ratio (2) The shape is not a triangle A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

E. 1 and 2 together are not sufficient

Step-by-step solution

Understanding Shape Similarity

For two shapes to be similar, their corresponding angles must be equal, and the lengths of corresponding sides must be in proportion (i.e., same ratio).

Evaluate Statement 1

Statement 1 says all pairs of corresponding sides are in the same ratio. If this is true, it means the condition for side proportionality is met. For most polygons, this would generally imply similarity, but we must consider if angle equality is guaranteed.

Consider Non-Triangular Shapes

Statement 2 says the shape is not a triangle. Triangular shapes only need the corresponding sides in the same ratio to be similar due to the AA (angle-angle) postulate. For non-triangular shapes, the equality of corresponding angles must be explicitly checked.

Assess Combined Conditions

When considering both conditions together: even though all sides are in the same ratio, knowing the shape is not a triangle does not provide sufficient information about the angles. Thus, we cannot confirm similarity without knowing about the angles of the non-triangular shape.

Conclusion

Given that neither condition alone nor the combination provides certainty about angle equality for non-triangular shapes, we cannot definitively determine similarity.

Key concepts

shape similarity

Understanding shape similarity is crucial for GMAT geometry problem solving. Two shapes are similar if they have the same shape but different sizes. In mathematical terms, this means that the corresponding angles of the two shapes are equal, and the corresponding sides are proportional. If two shapes meet these criteria, then the shapes are similar.

An easy way to think about this is to imagine scaling an image on your computer. Even though the size of the image changes, the shape remains the same, resulting in similar geometries.

corresponding sides ratio

The corresponding sides ratio is a fundamental concept in determining shape similarity. This ratio refers to the proportionate lengths of corresponding sides of two shapes. For shapes to be similar, their corresponding sides need to be in the same ratio.

For example, if a side of one shape is twice as long as the corresponding side in another shape, this must be true for all corresponding sides. Mathematically, if ewlineshape 1 has sides a, b, c, and shape 2 has sideska, kb, kc, then the corresponding sides form the same ratio ewlineewline ewlineewlineewlineewlinefor instance, k=3, ewlinea/ka = b/kb = c/kc = 1/3.

corresponding angles equality

Corresponding angles equality is another key principle in determining whether shapes are similar. It states that for two shapes to be similar, their corresponding angles must be equal. Simply having the ratio of the side lengths proportional is not enough.

This concept can be tricky in non-triangular shapes. For instance, in the GMAT problem, even if the sides are in the same ratio, without information about angle equality, we can't conclude the shapes are similar. Always check both side ratios and angle equality to be sure.

non-triangular shapes

Non-triangular shapes require extra caution when determining similarity. For triangles, if the sides are proportional, the angles are automatically equal because of the AA (angle-angle) postulate. Non-triangular shapes, like quadrilaterals or polygons, are more complex.

Just knowing that corresponding sides are in the same ratio is not sufficient to prove similarity in these cases; corresponding angles must also be equal. Therefore, always look out for specific evidence or data about angles when dealing with non-triangular shapes.

GMAT test strategies

For tackling GMAT geometry problems, it's vital to use effective test strategies. Here are a few tips:

• Always start by identifying known properties of shapes mentioned in the question.
• Break down the problem into smaller parts to understand the relationships between sides and angles.
• Draw diagrams whenever possible to visualize the problem better.
• Remember key geometry principles, like shape similarity, corresponding sides ratio, and angles equality.
• Don't rush. Use logic and systematically check all conditions provided in the question.

Approaching problems systematically helps reduce errors and boosts confidence during the test.