Right Triangle Sides

The problem asks us to find the area of a right triangle where the lengths of the sides are given as expressions involving '$x$': $(x - 5)$ cm, $(x - 10)$ cm, and $x$ cm. We need to use the properties of a right triangle to solve this.

Pythagorean Theorem Application

In a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (legs). This is known as the Pythagorean theorem: $a^2 + b^2 = c^2$.

First, we need to identify the hypotenuse. Since side lengths must be positive, we require:

• $(x - 10) > 0 \implies x > 10$
• $(x - 5) > 0 \implies x > 5$
• $x > 0$

The condition $x > 10$ ensures all sides are positive. Comparing the side lengths $(x - 5)$, $(x - 10)$, and $x$, the side '$x$' is the longest, so it must be the hypotenuse.

The other two sides, $(x - 5)$ and $(x - 10)$, are the legs.

Now, we apply the Pythagorean theorem:

$$ (x - 10)^2 + (x - 5)^2 = x^2 $$

Let's expand the terms:

$$ (x^2 - 2 \cdot x \cdot 10 + 10^2) + (x^2 - 2 \cdot x \cdot 5 + 5^2) = x^2 $$

$$ (x^2 - 20x + 100) + (x^2 - 10x + 25) = x^2 $$

Combine like terms on the left side:

$$ 2x^2 - 30x + 125 = x^2 $$

To solve for '$x$', we rearrange the equation into a standard quadratic form ($ax^2 + bx + c = 0$):

$$ 2x^2 - x^2 - 30x + 125 = 0 $$

$$ x^2 - 30x + 125 = 0 $$

Solving Quadratic Equation

We can solve this quadratic equation by factoring. We look for two numbers that multiply to 125 and add up to -30. These numbers are -5 and -25.

So, we can factor the equation as:

$$ (x - 5)(x - 25) = 0 $$

This gives two possible solutions for '$x$':

• $x - 5 = 0 \implies x = 5$
• $x - 25 = 0 \implies x = 25$

We must check which value of '$x$' is valid for the triangle's side lengths. We previously determined that '$x$' must be greater than 10 ($x > 10$).

• If $x = 5$, the side lengths would be $(5 - 5) = 0$ cm, $(5 - 10) = -5$ cm, and $5$ cm. These are not valid lengths for a triangle because one side is zero and another is negative.
• If $x = 25$, the side lengths are $(25 - 5) = 20$ cm, $(25 - 10) = 15$ cm, and $25$ cm. All these lengths are positive and valid.

Let's verify these sides ($15, 20, 25$) using the Pythagorean theorem:

So, the correct value for '$x$' is $25$.

Calculate Triangle Area

The area of a right triangle is given by the formula:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

In this case, the base and height are the two legs of the triangle, which are $(x - 10)$ cm and $(x - 5)$ cm.

Substitute the valid value $x = 25$ into the side lengths:

• Leg 1 = $(25 - 10) = 15$ cm
• Leg 2 = $(25 - 5) = 20$ cm

Now, calculate the area:

$$ \text{Area} = \frac{1}{2} \times 15 \text{ cm} \times 20 \text{ cm} $$

$$ \text{Area} = \frac{1}{2} \times 300 \text{ cm}^2 $$

$$ \text{Area} = 150 \text{ cm}^2 $$

Final Area Calculation

The area of the right triangle is 150 cm$^2$. This corresponds to Option 1.