The problem provided describes a right-angled triangle with a hypotenuse measuring 5 meters. You need to find the lengths of the other two sides, which are in the given ratio of 3:4.

Let's solve this step-by-step:

Step 1: Understanding the problem

• The triangle is a right-angled triangle.
• The hypotenuse (the side opposite the right angle) is 5 meters.
• The two other sides are in a 3:4 ratio.

Step 2: Use the ratio to express the sides

Let the lengths of the other two sides be $3x$ and $4x$, where $x$ is a common multiplier.

Step 3: Apply the Pythagorean theorem

The Pythagorean theorem states that in a right-angled triangle: $\text{(Hypotenuse)}^2 = \text{(Side 1)}^2 + \text{(Side 2)}^2$ Substitute the given values: $5^2 = (3x)^2 + (4x)^2$

Step 4: Simplify the equation

$25 = 9x^2 + 16x^2$ $25 = 25x^2$

Step 5: Solve for $x$

$x^2 = \frac{25}{25} = 1$ $x = 1$

Step 6: Find the lengths of the sides

Since $x = 1$, the lengths of the other two sides are:

• $3x = 3(1) = 3$ meters
• $4x = 4(1) = 4$ meters

Final Answer:

The lengths of the other two sides are 3 meters and 4 meters.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How does the Pythagorean theorem apply to non-right triangles?
2. Can the Pythagorean theorem be used to find the area of a triangle?
3. What happens if the ratio between the two sides was different?
4. How can trigonometric functions help in solving right triangles?
5. Can we solve the triangle using the sine and cosine rules?

Tip: Always ensure that the side lengths satisfy the Pythagorean theorem when dealing with right-angled triangles, especially when using ratios.