what is the perimeter of a right triangle whose area is 250 and one on its leg is 30?

To find the perimeter of the right triangle, we need to determine the lengths of the other two sides. Here’s how you can solve it:

### Step 1: Use the area formula
The area of a right triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2
\]
Given that the area is 250 and one leg (\(\text{leg}_1\)) is 30, we can solve for the other leg (\(\text{leg}_2\)):

\[
250 = \frac{1}{2} \times 30 \times \text{leg}_2
\]

\[
250 = 15 \times \text{leg}_2
\]

\[
\text{leg}_2 = \frac{250}{15} = \frac{50}{3} \approx 16.67
\]

### Step 2: Find the hypotenuse
Use the Pythagorean theorem to find the hypotenuse (\(c\)):

\[
c = \sqrt{(\text{leg}_1)^2 + (\text{leg}_2)^2}
\]

\[
c = \sqrt{30^2 + \left(\frac{50}{3}\right)^2}
\]

\[
c = \sqrt{900 + \frac{2500}{9}} = \sqrt{\frac{8100 + 2500}{9}} = \sqrt{\frac{10600}{9}} = \frac{\sqrt{10600}}{3}
\]

\[
c \approx \frac{103.027}{3} \approx 34.34
\]

### Step 3: Calculate the perimeter
The perimeter \(P\) of the triangle is the sum of all its sides:

\[
P = \text{leg}_1 + \text{leg}_2 + c
\]

\[
P = 30 + \frac{50}{3} + \frac{\sqrt{10600}}{3} \approx 30 + 16.67 + 34.34 \approx 81.01
\]

Thus, the perimeter of the triangle is approximately **81.01 units**.

Would you like further details or have any questions?

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**Related Questions:**

1. How do you calculate the hypotenuse if both legs are given?
2. What is the perimeter of a right triangle with legs 5 and 12?
3. How do you find the area of a triangle if all sides are known?
4. Can the area formula be used for any triangle, not just a right triangle?
5. How does the Pythagorean theorem work in non-right triangles?
6. How can you find the height of a right triangle given the area and one leg?
7. What are the properties of right triangles with specific angles, like 45-45-90?
8. How do trigonometric functions relate to right triangles?

**Tip:** Always double-check your calculations, especially when dealing with square roots and approximations.

Learn how to calculate the perimeter of a right triangle with an area of 250 square units and one leg measuring 30 units. Detailed step-by-step solution provided, covering key geometry concepts including the Pythagorean theorem and triangle perimeter calculation. Suitable for students in Grades 7-9.

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