Given the updated problem:

• The height of the larger triangle is 10.
• The height of the smaller triangle is 7.
• The base of the smaller triangle is 5.
• We need to solve for the base of the larger triangle.

Step 1: Identify the Triangles

We have two similar right triangles:

1. Smaller Triangle: Height = 7, Base = 5.
2. Larger Triangle: Height = 10, Base = $x$ (which we need to find).

Step 2: Set Up the Proportion

Since the triangles are similar, the ratios of their corresponding sides are equal:

$\frac{\text{Height of smaller triangle}}{\text{Base of smaller triangle}} = \frac{\text{Height of larger triangle}}{\text{Base of larger triangle}}$

Using the given values:

$\frac{7}{5} = \frac{10}{x}$

Step 3: Solve for $x$

Cross-multiply to solve for $x$:

$7x = 5 \times 10$

Simplify:

$7x = 50$

Now, divide both sides by 7:

$x = \frac{50}{7} \approx 7.14$

Final Answer

The base of the larger triangle is approximately $7.14$ units.

Would you like further details or have any additional questions? Here are some related questions you might be interested in:

1. How do you verify the similarity of triangles using side ratios?
2. What is the significance of similar triangles in geometry?
3. Can you apply similar triangles to find unknown heights or distances in real life?
4. How do you find the hypotenuse in a right triangle when given the other two sides?
5. How does scaling affect the perimeter and area of similar triangles?
6. What are the other methods for proving triangles are similar?
7. How can you use the concept of proportions in solving word problems?
8. What happens to the angles in similar triangles?

Tip: When working with proportions, double-check your setup by ensuring that the corresponding sides of the triangles are correctly matched.