Certainly! Let's address the last question regarding the right-angled triangle with a hypotenuse of 10 cm and the other two sides in a ratio of 3:4.
### Given:
- Hypotenuse (c): 10 cm
- Ratio of the other two sides (a:b): 3:4
### Find the lengths of the other two sides of the triangle.
### Solution:
1. Ratios and Variables:
- Let the two sides be [tex]\(3x\)[/tex] and [tex]\(4x\)[/tex] where [tex]\(x\)[/tex] is a common factor.
2. Using the Pythagorean Theorem:
- According to the Pythagorean theorem for a right-angled triangle:
[tex]\[
(3x)^2 + (4x)^2 = 10^2
\][/tex]
3. Calculating:
- Expanding the equation:
[tex]\[
9x^2 + 16x^2 = 100
\][/tex]
- Combining like terms:
[tex]\[
25x^2 = 100
\][/tex]
- Solving for [tex]\(x^2\)[/tex]:
[tex]\[
x^2 = 4
\][/tex]
- Taking the square root of both sides:
[tex]\[
x = 2
\][/tex]
4. Finding the Side Lengths:
- Using [tex]\(x = 2\)[/tex], calculate the lengths of the sides:
[tex]\[
\text{Side } a = 3x = 3 \times 2 = 6 \text{ cm}
\][/tex]
[tex]\[
\text{Side } b = 4x = 4 \times 2 = 8 \text{ cm}
\][/tex]
### Conclusion:
The lengths of the other two sides of the right-angled triangle are 6 cm and 8 cm, respectively.
