Problem 45 - Precalculus: Functions and Graphs

Question

The point on a coordinate line corresponding to \(\sqrt{2}\) may be determined by constructing a right triangle with sides of length \(1,\) as shown in the figure. Determine the points that correspond to \(\sqrt{3}\) and \(\sqrt{5},\) respectively. (Hint: Use the Pythagorean theorem.) CAN'T COPY THE GRAPH

Step-by-Step Solution

Answer

Use right triangles with sides 1 and \( \sqrt{2} \) for \( \sqrt{3} \), and sides 2 and 1 for \( \sqrt{5} \); plot these values on the line.

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is written as: \[ c^2 = a^2 + b^2 \].

Step 2: Construct a Right Triangle for \( \sqrt{3} \)

To find \( \sqrt{3} \), construct a right triangle where one side is 1 (like the previous triangle for \( \sqrt{2} \)), and another side is \( \sqrt{2} \) (the hypotenuse from the previous triangle). Apply the theorem: \[ h^2 = 1^2 + (\sqrt{2})^2 = 1 + 2 = 3 \]. Thus, \( h = \sqrt{3} \).

Step 3: Construct a Right Triangle for \( \sqrt{5} \)

For \( \sqrt{5} \), construct a right triangle with one side \( 2 \) (a longer side) and the other side \( 1 \). Apply the theorem: \[ h^2 = 2^2 + 1^2 = 4 + 1 = 5 \]. Therefore, the hypotenuse \( h = \sqrt{5} \).

Step 4: Plot the Points on the Coordinate Line

Once you have determined the lengths of \( \sqrt{3} \) and \( \sqrt{5} \), these values can be plotted on a number line: place \( \sqrt{3} \) between \( 1 \) and \( 2 \), and \( \sqrt{5} \) between \( 2 \) and \( 3 \).

Key Concepts

Coordinate Geometry

Coordinate geometry is a fascinating branch of mathematics that links algebra and geometry by using coordinates to specify the positions of points. In any coordinate system, each point is defined by a set of numbers. Typically, in two dimensions, these numbers are called the x-coordinate and the y-coordinate, representing the horizontal and vertical positions, respectively.
When dealing with Pythagorean theorem in coordinate geometry, you can locate points on a coordinate line or plane using the lengths of triangle sides as references. For instance, the theorem helps us determine the distance to which a point corresponding to a specific radical value should be plotted.
In our task, we have used right triangles to determine the positions of \( \sqrt{2} \), \( \sqrt{3} \), and \( \sqrt{5} \) on a coordinate line. By using specific lengths of triangle sides derived from the Pythagorean theorem, we can accurately assign these radical expressions to precise points. This approach is instrumental in visualizing radical values within a geometric framework.

Right Triangle Construction

Constructing right triangles is pivotal when applying the Pythagorean theorem, especially in coordinate geometry tasks. A right triangle has one angle measuring 90 degrees, with the sides opposite and adjacent to this right angle, referred to as the legs, while the side opposite the right angle is the hypotenuse - the longest side.
Let's explore the construction of right triangles for \( \sqrt{3} \) and \( \sqrt{5} \):

• For \( \sqrt{3} \), start with an existing triangle that has one leg of length 1 and a hypotenuse from a previous calculation (here, \( \sqrt{2} \)). Creating a fresh right triangle with one side of length 1 and a hypotenuse \( \sqrt{2} \) gives a new hypotenuse length that equals \( \sqrt{3} \).
• For \( \sqrt{5} \), use one leg measuring 2 and another measuring 1. From these dimensions, apply the Pythagorean theorem, yielding a hypotenuse equating to \( \sqrt{5} \).

By constructing these triangles, we translate abstract radical values into tangible lengths, allowing us to plot these points confidently on a coordinate line.

Radical Expressions

Radical expressions are expressions that contain roots, commonly square roots. Understanding radical expressions is crucial when plotting points such as \( \sqrt{3} \) and \( \sqrt{5} \) on a number line as seen in coordinate geometry.
These expressions often arise in geometry, particularly when using the Pythagorean theorem to determine triangle side lengths. In our exercise, by systematically applying the theorem, we calculated the square roots of integer sums to obtain radical expressions representing side lengths of right triangles.
For instance:

• To plot \( \sqrt{3} \), the task involves creating a triangle that naturally produces this radical via its hypotenuse, with previously known sides (\