Certainly! Let's solve this problem step-by-step:
### (i) Expressing the Sides in Terms of the Hypotenuse
Given that the sides of a right-angled triangle are less than the hypotenuse by 5 cm and 10 cm respectively:
- Let the hypotenuse be [tex]\( h \)[/tex].
- One side is then [tex]\( h - 5 \)[/tex].
- The other side is [tex]\( h - 10 \)[/tex].
### (ii) Finding the Value of the Hypotenuse
In a right-angled triangle, the Pythagorean theorem states that the square of the hypotenuse ([tex]\( h \)[/tex]) is equal to the sum of the squares of the other two sides. Applying this theorem:
[tex]\[ (h - 5)^2 + (h - 10)^2 = h^2 \][/tex]
Expanding and simplifying the equation:
[tex]\[ (h - 5)^2 = h^2 - 10h + 25 \][/tex]
[tex]\[ (h - 10)^2 = h^2 - 20h + 100 \][/tex]
So the equation becomes:
[tex]\[ (h^2 - 10h + 25) + (h^2 - 20h + 100) = h^2 \][/tex]
Combine and simplify the terms:
[tex]\[ h^2 - 10h + 25 + h^2 - 20h + 100 = h^2 \][/tex]
[tex]\[ 2h^2 - 30h + 125 = h^2 \][/tex]
Subtract [tex]\( h^2 \)[/tex] from both sides:
[tex]\[ h^2 - 30h + 125 = 0 \][/tex]
This is a quadratic equation in terms of [tex]\( h \)[/tex]. Solving this quadratic equation:
[tex]\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -30 \)[/tex], and [tex]\( c = 125 \)[/tex]. Substituting these values into the quadratic formula:
[tex]\[ h = \frac{30 \pm \sqrt{900 - 500}}{2} \][/tex]
[tex]\[ h = \frac{30 \pm \sqrt{400}}{2} \][/tex]
[tex]\[ h = \frac{30 \pm 20}{2} \][/tex]
Which gives us two solutions:
[tex]\[ h = \frac{30 + 20}{2} = 25 \][/tex]
[tex]\[ h = \frac{30 - 20}{2} = 5 \][/tex]
Since the value of the hypotenuse cannot be 5 (which would make one of the sides [tex]\( h - 10 \)[/tex] a negative number, which is not physically possible), the hypotenuse must be 25 cm.
### (iii) Finding the Sides of the Right-Angled Triangle
Now that we know the hypotenuse [tex]\( h = 25 \)[/tex]:
- One side is [tex]\( 25 - 5 = 20 \)[/tex] cm.
- The other side is [tex]\( 25 - 10 = 15 \)[/tex] cm.
So, the right-angled triangle has:
- Hypotenuse [tex]\( h = 25 \)[/tex] cm
- One side [tex]\( = 20 \)[/tex] cm
- Other side [tex]\( = 15 \)[/tex] cm
Thus, the sides of the right-angled triangle are 15 cm, 20 cm, and 25 cm.
