The hypotenuse of a right-angled triangle is 25 meters. If one of the other two sides is 3/4 of the other, determine the lengths of the two sides.

Let's denote the lengths of the two sides of the right-angled triangle as \( a \) and \( b \), with \( a \) being the shorter side and \( b \) the longer side. Given that one side is \(\frac{3}{4}\) of the other, we can express:

\[
a = \frac{3}{4}b
\]

The hypotenuse \( c \) is given as 25 meters. According to the Pythagorean theorem:

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

Substitute the known values into the equation:

\[
\left(\frac{3}{4}b\right)^2 + b^2 = 25^2
\]

Simplify the equation:

\[
\left(\frac{9}{16}b^2\right) + b^2 = 625
\]

Combine like terms:

\[
\frac{9}{16}b^2 + \frac{16}{16}b^2 = 625
\]

\[
\frac{25}{16}b^2 = 625
\]

Multiply both sides by 16 to clear the fraction:

\[
25b^2 = 10000
\]

Divide by 25:

\[
b^2 = 400
\]

Take the square root of both sides:

\[
b = 20
\]

Now, find \( a \):

\[
a = \frac{3}{4}b = \frac{3}{4} \times 20 = 15
\]

### Solution

The lengths of the two sides of the triangle are:

- \( a = 15 \) meters (shorter side)
- \( b = 20 \) meters (longer side)

Would you like further details, or do you have any questions?

### Related Questions

1. What is the area of the triangle given the side lengths?
2. How can you determine the other two angles in the triangle?
3. If one side of a right triangle is twice the length of another, how do you find the hypotenuse?
4. How do the lengths of a right triangle's sides relate to the trigonometric ratios?
5. What are the properties of a 3-4-5 right triangle?

### Tip

Remember that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Learn how to determine the side lengths of a right-angled triangle where the hypotenuse is 25 meters, and one side is 3/4 the length of the other. This problem uses the Pythagorean theorem and is suitable for students in Grades 8-10.

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