https://file.aiquickdraw.com/mathbot/file/1726141662201mais9nsb.jpg

The image shows a right triangle with the following side lengths:

- One leg is labeled \( x \) cm.
- The other leg is labeled \( 3x \) cm.
- The hypotenuse is labeled \( 10 \) cm.

We can solve for \( x \) using the Pythagorean theorem, which states:

\[
a^2 + b^2 = c^2
\]

where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse.

Here:
- \( a = x \)
- \( b = 3x \)
- \( c = 10 \)

Now, substituting these values into the Pythagorean theorem:

\[
x^2 + (3x)^2 = 10^2
\]

Let's solve for \( x \):

\[
x^2 + 9x^2 = 100
\]
\[
10x^2 = 100
\]
\[
x^2 = 10
\]
\[
x = \sqrt{10}
\]

So, \( x = \sqrt{10} \), or approximately \( x \approx 3.16 \) cm.

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. What is the perimeter of the triangle when \( x = \sqrt{10} \)?
2. What is the area of the triangle in terms of \( x \)?
3. How would the hypotenuse change if \( x \) were doubled?
4. If the hypotenuse were increased to 20 cm, how would that affect the value of \( x \)?
5. How can you derive the relationship between the legs and hypotenuse using trigonometric functions?

**Tip**: In right triangles, the Pythagorean theorem is a reliable way to find missing sides when two sides are known.

Find the value of x in a right triangle where the legs are x cm and 3x cm, and the hypotenuse is 10 cm. This problem covers essential geometry concepts and the Pythagorean theorem. A step-by-step solution is provided for students in Grades 8-10.

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