Problem Breakdown:

We are given that both triangles A and B are right triangles with the following information:

• Area of Triangle A = 50 square units
• Area of Triangle B = 90 square units
• Triangle A's second leg = 20 units
• Both triangles share one leg of the same length

We are tasked with finding the length of the second leg of Triangle B.

Step-by-Step Solution:

1. Area Formula for Right Triangles:
   The area $A$ of a right triangle can be calculated using the formula:
   $A = \frac{1}{2} \times \text{base} \times \text{height}$ In this case, the base and height correspond to the two legs of the triangle.

2. For Triangle A: Let the shared leg be denoted as $x$ and the second leg of Triangle A be 20 units. The area of Triangle A is given as 50 square units, so we can set up the equation for the area: $\frac{1}{2} \times x \times 20 = 50$ Simplifying the equation: $x \times 20 = 100 \quad \Rightarrow \quad x = \frac{100}{20} = 5$ So, the shared leg length is 5 units.

3. For Triangle B: Now, we use the same formula for Triangle B, where the area is given as 90 square units, and the shared leg is 5 units. Let the second leg of Triangle B be $y$. The area of Triangle B is: $\frac{1}{2} \times 5 \times y = 90$ Simplifying the equation: $5 \times y = 180 \quad \Rightarrow \quad y = \frac{180}{5} = 36$ Thus, the second leg of Triangle B is 36 units long.

Final Answer:

The second leg of Triangle B is 36 units.
Thus, the correct answer is E. 36.

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Would you like further details on how this calculation was performed? Let me know if you have any questions.

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Related Questions:

1. How would the area change if the second leg of Triangle A was a different length?
2. What if the shared leg was different between both triangles, how would the areas change?
3. Can the area of a right triangle ever be negative? Why or why not?
4. How does the Pythagorean theorem relate to the lengths of the legs in these triangles?
5. How do you calculate the hypotenuse of a right triangle when the legs are known?

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Tip: Always check the consistency of the units used (such as square units for area) to avoid errors in calculations.