Question

The altitude of a right triangle is 35 cm less than its base. If the hypotenuse is 65 cm, find the other two sides.

Hint:

In a right-angled triangle, the relationship between the base ($b$), altitude ($a$), and hypotenuse ($h$) is given by the Pythagoras theorem: $a^2 + b^2 = h^2$.

Answer 1

Step 1: Assume the base and altitude.
Let the base of the right triangle be \(x\) cm.
According to the question, the altitude is 35 cm less than the base.
So, altitude \(= x - 35\) cm.
The hypotenuse is given as 65 cm.

Step 2: Use the Pythagoras theorem.
For a right triangle:
\( (\text{base})^2 + (\text{altitude})^2 = (\text{hypotenuse})^2 \)
Substitute the values:
\(x^2 + (x - 35)^2 = 65^2\)

Step 3: Expand the equation.
\(x^2 + (x^2 - 70x + 1225) = 4225\)
\(2x^2 - 70x + 1225 = 4225\)

Step 4: Simplify the equation.
\(2x^2 - 70x + 1225 - 4225 = 0\)
\(2x^2 - 70x - 3000 = 0\)
Divide the equation by 2:
\(x^2 - 35x - 1500 = 0\)

Step 5: Solve the quadratic equation.
Factorize:
\(x^2 - 35x - 1500 = 0\)
\((x - 60)(x + 25) = 0\)
So,
\(x = 60\) or \(x = -25\).
Since length cannot be negative, we take x = 60.

Step 6: Find the altitude.
Altitude \(= x - 35\)
\(= 60 - 35\)
\(= 25\) cm.

Final Answer:
Base \(= 60\) cm
Altitude \(= 25\) cm
Hypotenuse \(= 65\) cm.