We are given that triangle $L$ is a scaled copy of triangle $K$, which means the two triangles are similar. Therefore, the sides of triangle $L$ are proportional to the sides of triangle $K$.

The sides of triangle $K$ are given as:

• One side is $\frac{3}{5}$
• Another side is $\frac{4}{5}$

For triangle $L$, one corresponding side is $\frac{9}{10}$, and we are tasked with finding the value of $j$, the other side in triangle $L$, corresponding to the $\frac{4}{5}$ side of triangle $K$.

Step 1: Find the scale factor

The ratio between the corresponding sides of triangles $L$ and $K$ can be found using the side that is given for both: $\text{Scale factor} = \frac{\frac{9}{10}}{\frac{3}{5}} = \frac{9}{10} \times \frac{5}{3} = \frac{9 \times 5}{10 \times 3} = \frac{45}{30} = 1.5$

Step 2: Use the scale factor to find $j$

Now, we apply this scale factor to the side $\frac{4}{5}$ in triangle $K$: $j = 1.5 \times \frac{4}{5} = \frac{1.5 \times 4}{5} = \frac{6}{5} = 1.2$

Final Answer:

The value of $j$ is $1.2$.

Would you like more details or have any further questions?

Here are 5 related questions to consider:

1. How do you determine whether two triangles are similar?
2. What happens to the angles of similar triangles?
3. If a triangle is scaled by a factor of 2, how do the areas of the two triangles compare?
4. Can two triangles be similar if their corresponding sides do not have the same ratio?
5. How do you find the length of a missing side in a non-right triangle using proportions?

Tip: When dealing with similar triangles, always check the corresponding sides to ensure they are proportional!