The longest side of a right angled triangle is 4cm longer than one side and 2 cm longer than the other side. Find the longest side.

Answer

To solve the problem, we need to find the longest side of a right-angled triangle, which is the hypotenuse. Let's denote the sides of the triangle as follows:

- Let \( H \) be the length of the hypotenuse (the longest side).
- Let \( B \) be the length of one side (base).
- Let \( P \) be the length of the other side (perpendicular).

According to the problem:
1. The hypotenuse \( H \) is 4 cm longer than the base \( B \).
2. The hypotenuse \( H \) is also 2 cm longer than the perpendicular \( P \).

From these statements, we can set up the following equations:

1. \( H = B + 4 \) (Equation 1)
2. \( H = P + 2 \) (Equation 2)

Now, we can express \( B \) and \( P \) in terms of \( H \):

From Equation 1:
\[ B = H - 4 \] (Equation 3)

From Equation 2:
\[ P = H - 2 \] (Equation 4)

Next, we will use the Pythagorean theorem, which states that in a right-angled triangle:
\[ H^2 = B^2 + P^2 \]

Substituting Equations 3 and 4 into the Pythagorean theorem gives us:
\[ H^2 = (H - 4)^2 + (H - 2)^2 \]

Now, let's expand the right side:
1. \( (H - 4)^2 = H^2 - 8H + 16 \)
2. \( (H - 2)^2 = H^2 - 4H + 4 \)

Combining these:
\[ H^2 = (H^2 - 8H + 16) + (H^2 - 4H + 4) \]
\[ H^2 = H^2 + H^2 - 12H + 20 \]
\[ H^2 = 2H^2 - 12H + 20 \]

Now, let's rearrange the equation:
\[ 0 = 2H^2 - H^2 - 12H + 20 \]
\[ 0 = H^2 - 12H + 20 \]

This is a quadratic equation in the standard form:
\[ H^2 - 12H + 20 = 0 \]

Next, we can solve this quadratic equation using the factorization method or the quadratic formula. Let's factor it:
We need two numbers that multiply to \( 20 \) and add up to \( -12 \). The numbers are \( -10 \) and \( -2 \):
\[ (H - 10)(H - 2) = 0 \]

Setting each factor to zero gives us:
1. \( H - 10 = 0 \) → \( H = 10 \)
2. \( H - 2 = 0 \) → \( H = 2 \)

Since \( H \) represents the length of the hypotenuse, it cannot be 2 cm (as it must be the longest side). Therefore, we have:
\[ H = 10 \, \text{cm} \]

Now, we can find the lengths of the other sides:
Using Equation 3:
\[ B = H - 4 = 10 - 4 = 6 \, \text{cm} \]

Using Equation 4:
\[ P = H - 2 = 10 - 2 = 8 \, \text{cm} \]

Thus, the lengths of the sides of the triangle are:
- Hypotenuse \( H = 10 \, \text{cm} \)
- Base \( B = 6 \, \text{cm} \)
- Perpendicular \( P = 8 \, \text{cm} \)

**Final Answer: The longest side (hypotenuse) is 10 cm.**