The altitude of a right-angled triangle is 7cm less than its base. If the hypotenuse is 13cm, find the other two sides.

Let's solve this step by step.

We are given a right-angled triangle with:

The hypotenuse = 13 cm
The altitude = 7 cm less than the base
Let:

The base of the triangle be "x" cm.
The altitude (which is perpendicular to the base) be "x - 7" cm.
Now, apply the Pythagorean theorem to the right-angled triangle. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides (base and altitude). Therefore:

x² + (x - 7)² = 13²

Now, expand the equation:

x² + (x² - 14x + 49) = 169

Simplify:

2x² - 14x + 49 = 169

Subtract 169 from both sides:

2x² - 14x + 49 - 169 = 0

2x² - 14x - 120 = 0

Now, divide the whole equation by 2 to simplify:

x² - 7x - 60 = 0

This is a quadratic equation. To solve it, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

For the equation x² - 7x - 60 = 0, the coefficients are: a = 1, b = -7, and c = -60.

Now, substitute these values into the quadratic formula:

x = (7 ± √((-7)² - 4(1)(-60))) / 2(1) x = (7 ± √(49 + 240)) / 2 x = (7 ± √289) / 2 x = (7 ± 17) / 2

Now solve for the two possible values of x:

x = (7 + 17) / 2 = 24 / 2 = 12
x = (7 - 17) / 2 = -10 / 2 = -5
Since a length cannot be negative, we discard x = -5, so the base of the triangle is 12 cm.

Now, the altitude is x - 7, so:

Altitude = 12 - 7 = 5 cm.

Therefore, the other two sides of the triangle are:

Base = 12 cm
Altitude = 5 cm