Question

In right triangle \(A B C\) above, side \(A B\) has a length of \(x\) units, side \(A C\) has a length of 12 units, and side \(B C\) has a length of \(x\) +8 units. What is the area of \(A B C\) in square units? \(\square\)square units

Short answer

The area of triangle ABC is 30 square units.

Step-by-step solution

Apply Pythagorean Theorem

Use the Pythagorean theorem in right triangle \(ABC\), where \(A C \) is one leg, \(A B \) is the other leg, and \(B C \) is the hypotenuse. The Pythagorean theorem states \((A B)^2 + (A C)^2 = (B C)^2 \).

Write the equations

Since \(A B = x\), \(A C = 12\), and \(B C = x + 8\), substitute these values into the equation: \(x^2 + 12^2 = (x + 8)^2\).

Expand and simplify

Expand the equation to get: \(x^2 + 144 = x^2 + 16x + 64\). Then simplify by subtracting \(x^2\) from both sides: \(144 = 16x + 64\).

Solve for x

Isolate \(x\) by subtracting 64 from both sides: \(144 - 64 = 16x\). This simplifies to \(80 = 16x\). Divide both sides by 16: \(\frac{80}{16} = x \). Hence, \(x = 5\).

Calculate the area of the triangle

The area \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). In this case, \(A B = 5 \) (base) and \(A C = 12 \) (height). Substitute these values into the area formula: \(\text{Area} = \frac{1}{2} \times 5 \times 12 \).

Finalize the area calculation

Simplify to find the area: \(\text{Area} = \frac{1}{2} \times 60 = 30 \) square units.

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring 90 degrees. This special angle makes it easier to work with through specific mathematical rules and properties.
One important property is that the side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides are the legs of the triangle.
In a right triangle, you can always use the Pythagorean Theorem to find the lengths of sides. This theorem states that for any right triangle with legs of length \(a\) and \(b\), and hypotenuse of length \(c\), the formula \[ a^2 + b^2 = c^2 \] always holds true.
In our exercise, triangle ABC has side lengths \(AB\), \(AC\), and \(BC\), where \(BC\) is the hypotenuse.

Area Calculation

The area of a triangle can be calculated if you know the base and height of the triangle. The formula is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In a right triangle, one leg can be considered the base and the other leg can be considered the height.
For our problem, we have that \(AB = 5 \) units and \(AC = 12 \) units. We substitute these values into the area formula to find the total area of triangle ABC:
\[ \text{Area} = \frac{1}{2} \times 5 \times 12 \]
When simplified, it results in \[ \text{Area} = 30 \text{ square units} \].
This formula is especially useful in different geometric and real-life applications involving right triangles.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve for unknown variables.
In our exercise, we apply algebraic manipulation to solve for \(x\).
The Pythagorean equation we start with is: \[ x^2 + 12^2 = (x + 8)^2 \]
First, we expand the equation: \[ x^2 + 144 = x^2 + 16x + 64 \]
By subtracting \(x^2\) from both sides, we get: \[ 144 = 16x + 64 \]
Next, we isolate \(x\) by subtracting 64 from both sides: \[ 80 = 16x \]
Finally, solve for \(x\) by dividing both sides by 16: \[ x = 5 \]
Understanding these steps ensures a strong foundation in algebra, which is crucial for tackling more math problems in the future.