91Ó°ÊÓ

The distance formula is a crucial tool in geometry, particularly when you're dealing with coordinates on a plane. This formula allows you to calculate the distance between two points. If you have two points, say \(A = (x_1, y_1)\) and \(B = (x_2, y_2)\), the distance \(d\) between them can be calculated using the formula: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \] This formula is derived from the Pythagorean theorem (which we'll explore later), by considering the line between the two points as the hypotenuse of a right triangle.
In our exercise, we used the distance formula to determine the lengths of the sides of the triangle with vertices \(P=(0,0), Q=(0,3),\) and \(R=(4,3)\). For example, to find the length of side \(PQ\) you calculate: \[ PQ = \sqrt{(0-0)^2 + (3-0)^2} = 3 \]
This straightforward approach reveals how the distance formula helps in determining side lengths accurately without any complex calculations.

The Pythagorean theorem is fundamental in trigonometry and relates to right triangles. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
So, in a triangle with sides \(a\), \(b\), and hypotenuse \(c\), the theorem is represented as: \[ a^2 + b^2 = c^2 \]
This theorem is not only useful for proving right triangle relationships but also for solving various geometric problems. In the given exercise, after computing the side lengths, we verified the triangle is right-angled using Pythagorean theorem: \[ PQ^2 + QR^2 = RP^2 \]\[ 3^2 + 4^2 = 5^2 \]
The left-hand side sums to 25, confirming that \(RP = 5\) matches as the hypotenuse. Such verification provides not only an answer but also confidence in the calculation process.

A right triangle is a type of triangle that has one angle equal to \(90^\circ\). Right triangles are fundamental in trigonometry because they allow us to define and understand trigonometric ratios for angles.

• The side opposite the right angle is called the hypotenuse and is the longest side in the triangle.
• The other two sides are known as the legs, which can be referred to as base and height depending on orientation.

In our exercise, the triangle \(PQR\) was identified as a right triangle with the right angle situated at point \(Q\). This makes \(RP\) the hypotenuse and sides \(PQ\) and \(QR\) the legs. Using this information, we not only calculated side lengths with the distance formula but also verified angles using trigonometric functions. The classification of triangles, like identifying \(PQR\) as a right triangle, helps in understanding geometric properties and making calculations more intuitive.