We are given that quadrilateral FGHI is similar to quadrilateral JKLM. When two figures are similar, their corresponding sides are proportional. This allows us to set up a proportion between the corresponding sides of the two quadrilaterals.

From the image:

• Side $FH$ in quadrilateral FGHI corresponds to side $ML$ in quadrilateral JKLM.
• Side $FG$ in quadrilateral FGHI corresponds to side $MK$ in quadrilateral JKLM.
• We are asked to find the length of side $MJ$ in quadrilateral JKLM.

Given:

• $FH = 3$ and $ML = 6$
• $FG = 4.5$ and $MK$ is unknown (but not needed for this problem).

We can use the ratio of the corresponding sides $FH$ and $ML$ to find $MJ$, knowing that the two quadrilaterals are similar.

Proportionality:

$\frac{FH}{ML} = \frac{FG}{MK} = \frac{FI}{MJ}$

Using the first ratio $\frac{FH}{ML}$: $\frac{3}{6} = \frac{FI}{MJ}$ Thus, $\frac{3}{6} = \frac{1}{2}$.

Now, side $FI = 4.5$, so using the proportion:

$\frac{1}{2} = \frac{4.5}{MJ}$

Solving for $MJ$:

$MJ = 4.5 \times 2 = 9$

Thus, the length of side $MJ$ is 9 units.

Would you like a further breakdown or have any questions?

Related Questions:

1. How can we find the length of side $MK$ in quadrilateral JKLM?
2. What would the area ratio of these similar quadrilaterals be?
3. If the perimeter of quadrilateral FGHI is known, how do we find the perimeter of JKLM?
4. Can we determine the angles of the quadrilaterals based on the given information?
5. How do similarity rules apply to other polygons besides quadrilaterals?

Tip:

When working with similar figures, always identify corresponding sides and angles first, as they provide the foundation for setting up proportions.