Question

Find the length of the longest side of the triangle formed by the line \(3 \mathrm{x}+4 \mathrm{y}=12\) with the coordinate axes. (1) 9 (2) 16 (3) 5 (4) 7

Short answer

Answer: 5

Step-by-step solution

Find the coordinates of the vertices

To find the coordinates of the vertices, we need the points where the line intersects the coordinate axes (x-axis and y-axis): 1. Intersection with the x-axis (y = 0): \(3x + 4(0) = 12\) \(3x = 12\) \(x = 4\) So, the first vertex is at point A(4, 0). 2. Intersection with the y-axis (x = 0): \(3(0) + 4y = 12\) \(4y = 12\) \(y = 3\) So, the second vertex is at point B(0, 3). The third vertex is the origin C(0, 0). Now we have the three vertices of the triangle: A(4, 0) B(0, 3) C(0, 0)

Use the distance formula to find the length of each side

We will use the distance formula \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the lengths of the sides AB, BC and CA. 1. Side AB: \(D_{AB} = \sqrt{(0 - 4)^2 + (3 - 0)^2}\) \(D_{AB} = \sqrt{(-4)^2 + (3)^2}\) \(D_{AB} = \sqrt{16 + 9}\) \(D_{AB} = \sqrt{25} = 5\) 2. Side BC: \(D_{BC} = \sqrt{(0 - 0)^2 + (3 - 0)^2}\) \(D_{BC} = \sqrt{(0)^2 + (3)^2}\) \(D_{BC} = \sqrt{0 + 9}\) \(D_{BC} = \sqrt{9} = 3\) 3. Side CA: \(D_{CA} = \sqrt{(0 - 4)^2 + (0 - 0)^2}\) \(D_{CA} = \sqrt{(-4)^2 + (0)^2}\) \(D_{CA} = \sqrt{16 + 0}\) \(D_{CA} = \sqrt{16} = 4\)

Determine the longest side

We have now calculated the lengths of the sides: Side AB: 5 Side BC: 3 Side CA: 4 The longest side is AB with a length of 5. Therefore, the answer is (3) 5.

Key concepts

Distance Formula

The distance formula is a handy mathematical tool to calculate the distance between two points in a plane. This is especially useful when you have the coordinates of these points.
To describe it, let's say you have two points,

• Point 1 with coordinates error: Please enter valid LaTeX code (possibly invalid LaTeX Error: Missing $ inserted), for instance, (x_1, y_1)
• Point 2 with coordinates (x_2, y_2)

The distance between these points is given by the formula: \[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is derived from the Pythagorean theorem, showing the power of geometry in solving real-world problems. By plugging in the coordinates of the points into this formula, you can find the length of any side of a triangle when given two of its vertices.
For the problem at hand, we used this formula to find the lengths of the sides of a triangle formed by the line with the coordinate axes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves solving geometrical problems using the coordinate plane. This combines algebra with geometry and provides a powerful way to analyze shapes and their properties.
Using coordinates simplifies many grips of geometric concepts. For example, to find where a line intersects the axes or determine the equation of a line, coordinate geometry comes into play. In essence, every point on the plane can be described using two numbers

• x-coordinate determines how far along the x-axis the point is
• y-coordinate shows the position along the y-axis

In our exercise, the intersection of the line with the x-axis and y-axis gave us vital points to form the triangle. Understanding coordinate geometry is crucial for dissecting and solving these kinds of problems
easily and with precision.

Intersection with Axes

Finding the intersection of a line with the axes involves a simple technique of setting the opposite coordinate to zero.
When a line meets the

• x-axis, it crosses at a point where the y-coordinate is zero
• y-axis, it crosses at a point where the x-coordinate is zero

This gives us systematic ways to determine these points using the equation of the line.
h4>The Steps to Find Intersections:

• For the x-axis, set \(y = 0\) in the equation and solve for \(x\).
• For the y-axis, set \(x = 0\) and solve for \(y\).

In our problem, solving these two simple equations allowed us to find the critical points where the line, given by the equation \(3x + 4y = 12\), met the coordinate axes. These intersections provided the vertices of the triangle necessary to solve the problem at hand.