The problem involves a right-angled triangle with sides labeled in terms of $x$. The lengths of the sides are as follows:

• One leg: $x + 5$,
• Another leg: $x - 2$,
• The hypotenuse: $2x - 1$.

We are tasked with finding the value of $x$ and the area of the triangle.

Step 1: Apply the Pythagorean theorem

For a right-angled triangle, the Pythagorean theorem states: $(\text{leg}_1)^2 + (\text{leg}_2)^2 = (\text{hypotenuse})^2$ Substituting the values: $(x + 5)^2 + (x - 2)^2 = (2x - 1)^2$

Step 2: Expand both sides

Expand each square:

• $(x + 5)^2 = x^2 + 10x + 25$,
• $(x - 2)^2 = x^2 - 4x + 4$,
• $(2x - 1)^2 = 4x^2 - 4x + 1$.

Now substitute back into the Pythagorean equation: $(x^2 + 10x + 25) + (x^2 - 4x + 4) = 4x^2 - 4x + 1$

Step 3: Simplify the equation

Combine like terms on the left side: $2x^2 + 6x + 29 = 4x^2 - 4x + 1$

Move all terms to one side: $0 = 4x^2 - 4x + 1 - 2x^2 - 6x - 29$ $0 = 2x^2 - 10x - 28$

Step 4: Solve the quadratic equation

Divide the equation by 2 to simplify: $x^2 - 5x - 14 = 0$

Use the quadratic formula to solve for $x$: $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}$ $x = \frac{5 \pm \sqrt{25 + 56}}{2}$ $x = \frac{5 \pm \sqrt{81}}{2}$ $x = \frac{5 \pm 9}{2}$ Thus, the two possible values for $x$ are: $x = \frac{5 + 9}{2} = 7 \quad \text{or} \quad x = \frac{5 - 9}{2} = -2$

Since $x$ must be positive for the side lengths to make sense, we have: $x = 7$

Step 5: Find the area of the triangle

The area of a right-angled triangle is given by: $\text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$ Substitute the values:

• $\text{leg}_1 = x + 5 = 7 + 5 = 12$,
• $\text{leg}_2 = x - 2 = 7 - 2 = 5$.

Thus, the area is: $\text{Area} = \frac{1}{2} \times 12 \times 5 = 30$

Final Answer:

• $x = 7$,
• The area of the triangle is $30 \, \text{square units}$.

Would you like more details on any specific step? Here are some related questions:

1. How is the quadratic formula derived?
2. What is the significance of the Pythagorean theorem in geometry?
3. Can you explain more examples of solving quadratic equations?
4. How do you find the area of different types of triangles?
5. What are the practical applications of right-angled triangles?

Tip: Always check that the values for the variable make sense in the context of the problem, especially when dealing with geometric shapes.