Question 5

We will use the Pythagorean theorem to form a quadratic equation.

Step 1 — Form the equation

Let's define the sides of the triangle. Let the base of the right triangle be $x$ cm. The altitude is $7$ cm less than the base. So, the altitude is $(x - 7)$ cm. The hypotenuse is given as $13$ cm. We use the Pythagorean theorem. The square of the hypotenuse equals the sum of the squares of the other two sides.

$x^2 + (x - 7)^2 = 13^2$

$x^2 + (x^2 - 14x + 49) = 169$

$2x^2 - 14x + 49 = 169$

$2x^2 - 14x + 49 - 169 = 0$

$2x^2 - 14x - 120 = 0$

Let's divide the entire equation by $2$.

$\boxed{x^2 - 7x - 60 = 0}$

Step 2 — Solve the equation

We have the quadratic equation $x^2 - 7x - 60 = 0$. Let's factorize this equation. We need two numbers that multiply to $-60$ and add up to $-7$. These numbers are $-12$ and $5$.

$x^2 - 12x + 5x - 60 = 0$

$x(x - 12) + 5(x - 12) = 0$

$(x - 12)(x + 5) = 0$

This gives us two possible values for $x$. Either $x - 12 = 0$ or $x + 5 = 0$. So, $x = 12$ or $x = -5$. A side length cannot be negative. Therefore, we reject $x = -5$. The base of the triangle is $x = \mathbf{12}$ cm. The altitude is $x - 7$. Altitude $= 12 - 7 = \mathbf{5}$ cm.

$\boxed{\text{The other two sides are 5 cm and 12 cm.}}$

Answer

The other two sides of the triangle are 5 cm and 12 cm.