Problem 27 - Algebra 2 and Trigonometry

Question

For each \(\triangle O R S, O\) is the origin, \(R\) is on the positive ray of the \(x\) -axis and \(\overline{P S}\) is the altitude from \(S\) to \(\overrightarrow{O R}\) . Find the exact coordinates of \(R\) and \(S .\) b. Find the exact area of \(\triangle O R S .\) \(O R=20, O S=R S, P S=10\)

Step-by-Step Solution

Answer

Point \(R\) has coordinates (20,0), point \(S\) is at (10,10), and the area of \(\triangle ORS\) is 100 square units.

Step 1: Define the coordinates of points

Let's start by placing point \(O\) at the origin. So, the coordinates of point \(O\) are \((0,0)\). Since \(R\) is on the positive ray of the \(x\)-axis, its coordinates can be given by \(R(20,0)\) because \(OR = 20\).

Step 2: Set up the equation for point S

Point \(S\) is such that \(OS = RS\) and \(PS\) is the altitude from \(S\) to \(OR\) with \(PS = 10\). Let \(S\) have coordinates \((x, y)\). Since \(PS = 10\) and is perpendicular to \(OR\), the line \(PS\) is vertical, so \(R\) lies directly below or above \(S\).Since \(RS = OS\), the points \(S\) lie on the circle centered at \(R\) with radius equal to the length of \(OR\). Thus, \(x^2 + y^2 = RS^2 = (20 - x)^2 + y^2\).

Step 3: Solve for the y-coordinate of S using altitude law

The altitude length \(PS\) from \(S\) to \(OR\) is \(y = 10\), consequently we have:\[y = 10\]for point \(S\). Knowing \(S\) forms a right triangle at \((20,0)\), and the \(y\)-coordinate is given directly by the altitude, which is the distance of \(S\) from \(OR\) line directly.

Step 4: Determine x-coordinate of S

Since \(PS = 10\), it is directly vertical from \(R(20,0)\) then \( RS = OS \) each being the same which denotes a 45-degree right angle triangle at \(S\).Given the condition \(OS = RS\), equate the two distances \[ OS = \sqrt{x^2 + 10^2} = RS = \sqrt{(20-x)^2 + 10^2} \]. Simplifying both sides, cancel the 100 from each ends,\[ x^2 = (20-x)^2 \]which implies:\[ x = 10 \text{ since it simplifies to } x = 10 \] evaluated respecting altitude condition.

Step 5: Find area of \(\triangle ORS\)

The area \(A\) of \(\triangle ORS\) can be calculated using the formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\). Here, \(OR = 20\) serves as the base, and \(PS = 10\) serves as the height:\[ A = \frac{1}{2} \times 20 \times 10 = 100 \].

Key Concepts

Coordinates in Geometry

In geometry, coordinates are used to pinpoint the exact location of points on a plane using an ordered pair of numbers. Each point is identified by coordinates in the format \((x, y)\). Here's a simple breakdown:

• The first number, called the x-coordinate, defines a point’s horizontal position along the x-axis.
• The second number, or y-coordinate, indicates a point’s vertical position along the y-axis.

When discussing triangles in geometry, knowing the coordinates of each vertex (corner) is crucial to further calculate distances, angles, and areas. In the exercise, the coordinates of point \(O\) are \((0, 0)\) because it is the origin of the plane. Point \(R\) lies on the x-axis at \(20\), which gives it coordinates \((20, 0)\).Understanding the placement of these points enables us to visualize and calculate various properties of the triangle formed by these points.

Calculating Triangle Area

The area of a triangle in geometry can often be calculated using the simple formula:\[ A = \frac{1}{2} \times \, \text{base} \, \times \, \text{height} \]In the case of Triangle \(\triangle ORS\), the base \(OR\) is clearly known, being a horizontal line on the x-axis stretching from point \(O(0, 0)\) to point \(R(20, 0)\), so the length of the base is \( 20 \).

• The given altitude is \(PS = 10\), which serves as the height of the triangle.
• It is crucial to ensure that the base and height are perpendicular. In this case, \(PS\) being vertical is naturally perpendicular to \(OR\).

By applying the formula, we plug in the values:\[ A = \frac{1}{2} \times 20 \times 10 = 100 \]This shows how the area formula translates into a numerical answer by understanding the positioning of the base and height on the Cartesian plane.

Altitude in Triangles

The altitude of a triangle is a line segment drawn from a vertex to the opposite side, becoming perpendicular to it in the process. This specific feature is quite useful for calculating areas as it represents the height of the triangle.In \(\triangle ORS\), the line segment \(PS\) acts as the altitude:

• \(PS\) starts from vertex \(S\) and is perpendicular to the base \(OR\).
• Its length, given as \(10\), directly measures the height of the triangle.

Identifying and using altitudes effectively requires understanding both the triangle's geometry and how perpendicularity plays a role:

• Perpendicular lines intersect at a \(90^\circ\) angle, which is an essential property for using in calculations.
• For right triangles, altitudes often coincide with one side, simplifying calculations further.

In this particular problem, recognizing \(PS\) as the altitude not only aids in finding the triangle's area but also clarifies the relationship and position of point \(S\) relative to points \(O\) and \(R\). This insight helps solve for the coordinates of \(S\).