The given diagram involves two similar triangles (as indicated by the equal angles). The task is to find the missing side ($?$) using proportions.

Step-by-step solution:

1. Identify corresponding sides of similar triangles:

   • The longer hypotenuse of the larger triangle corresponds to the longer hypotenuse of the smaller triangle.
   • The shorter base of the larger triangle corresponds to the shorter base of the smaller triangle.

   From the image:

   • Hypotenuse of the larger triangle: $21$
   • Hypotenuse of the smaller triangle: $8$
   • Base of the larger triangle: $20$
   • Base of the smaller triangle: $?$ (to be found).

2. Set up the proportion using similar triangles: $\frac{\text{Hypotenuse of larger triangle}}{\text{Hypotenuse of smaller triangle}} = \frac{\text{Base of larger triangle}}{\text{Base of smaller triangle}}$ Substituting the given values: $\frac{21}{8} = \frac{20}{?}$

3. Solve for the missing side ($?$): Cross-multiply: $21 \cdot ? = 20 \cdot 8$ $? = \frac{20 \cdot 8}{21}$ $? = \frac{160}{21}$ $? \approx 7.62 \, \text{units (rounded to two decimal places)}.$

Final Answer:

The length of the missing side ($?$) is approximately 7.62 units.

Would you like a more detailed explanation of the concepts involved?

Related Questions:

1. What are the properties of similar triangles, and how are proportions used to solve for unknowns?
2. How do you calculate the scale factor between two similar triangles?
3. What methods can be used to verify if two triangles are similar?
4. Can the Pythagorean theorem be used here to double-check results?
5. How would the solution change if a different pair of sides were involved?

Tip:

When working with similar triangles, always ensure that the ratios are set up between corresponding sides (matching sides from each triangle). This helps avoid errors in calculations!