Question

two triangles (pink labeled height, base and blue labeled height, base) are shown. four proportion options:

1. $\frac{\text{original height}}{\text{original base}} = \frac{\text{reduction height}}{\text{reduction base}}$
2. $\frac{\text{original height}}{\text{reduction base}} = \frac{\text{reduction height}}{\text{original base}}$
3. $\frac{\text{original height}}{\text{original base}} = \frac{\text{reduction base}}{\text{reduction height}}$
4. $\frac{\text{original height}}{\text{reduction base}} = \frac{\text{original base}}{\text{reduction height}}$

Explanation:

Step1: Identify Similar Triangles

The two triangles (original - pink, reduced - blue) are similar (same shape, different size). For similar triangles, the ratios of corresponding sides are equal.

Step2: Match Corresponding Sides

The height and base of the original triangle correspond to the height and base of the reduced triangle. So, $\frac{\text{original height}}{\text{original base}}$ (ratio of height to base for original) should equal $\frac{\text{reduction height}}{\text{reduction base}}$ (ratio of height to base for reduced), as corresponding side ratios in similar triangles are equal.

Answer:

$\boldsymbol{\frac{\text{original height}}{\text{original base}} = \frac{\text{reduction height}}{\text{reduction base}}}$ (the first option)