91Ó°ÊÓ

The distance formula helps us find the length of a line segment between two points in a plane. It's particularly handy in geometry, especially when dealing with right triangles. The formula is derived from the Pythagorean Theorem and is given by: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]

• Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• The difference between the \(x\)-coordinates is squared, as well as the difference between the \(y\)-coordinates.
• The sum of these squares gives us the length of the hypotenuse of a right-angled triangle formed by these differences.

Using this formula, we calculated the sides of the triangle in the problem to verify if it forms a right triangle. Applying the distance formula reveals lengths such as \(7\sqrt{2}\) and \(4\sqrt{2}\), which our calculations showed were essential for recognizing the triangle's nature.

The Pythagorean Theorem is a cornerstone in geometry, especially useful for determining if a triangle is right-angled. It states that for a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). Mathematically, it is expressed as: \[a^2 + b^2 = c^2\]

• Here, \(a\) and \(b\) are the lengths of the legs.
• \(c\) represents the length of the hypotenuse.

In our problem, the sides measured using the distance formula were verified using the Pythagorean Theorem.

Solving the Right Triangle

We checked if \((7\sqrt{2})^2 + (4\sqrt{2})^2 = 130\), finding that the equation holds true. This confirmed the presence of a right angle in the triangle, with \(CA\) as the hypotenuse. Understanding this theorem is crucial in identifying and confirming right triangles.

Calculating the area of a triangle requires an understanding of base and height. For right-angled triangles, this process is simplified, as the two shorter sides can be used directly as base and height. The area is calculated using the formula: \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]

• For the given triangle, \(7\sqrt{2}\) and \(4\sqrt{2}\) serve as base and height.
• By plugging these into the formula: \(\frac{1}{2} \times 7\sqrt{2} \times 4\sqrt{2}\), we simplified it to 28 square units.

The process of finding area is greatly facilitated for right triangles due to the perpendicular relationship between the base and the height, removing the need for more complex formulas. This makes right triangles not only easier to work with but also highlights the importance of understanding the relationship between the sides.