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: Understanding the Concept:
In a right-angled triangle, the relationship between the base ($b$), altitude ($a$), and hypotenuse ($h$) is given by the Pythagoras theorem: $a^2 + b^2 = h^2$.
Step 2: Setting up the Quadratic Equation:
Let the base be $x$ cm.
Then, altitude $= (x - 35)$ cm. Hypotenuse $= 65$ cm.
According to Pythagoras theorem:
$x^2 + (x - 35)^2 = 65^2$
$x^2 + (x^2 - 70x + 1225) = 4225$
$2x^2 - 70x + 1225 - 4225 = 0$
$2x^2 - 70x - 3000 = 0 \implies x^2 - 35x - 1500 = 0$
Step 3: Solving the Equation:
Splitting the middle term:
$x^2 - 60x + 25x - 1500 = 0$
$x(x - 60) + 25(x - 60) = 0$
$(x - 60)(x + 25) = 0$
Since side length cannot be negative, $x = 60$.
Base $= 60$ cm; Altitude $= 60 - 35 = 25$ cm.
Step 4: Final Answer:
The two sides are 60 cm and 25 cm.