Find the Area of a Triangle Formed by Line with Slope 3 and Coordinate Axes

Math Problem Statement

Line $\ell$ is drawn in the coordinate plane below. If the slope of line $\ell$ is $3,$ then find the area of the triangle determined by the line and the coordinate axes.

[asy] unitsize(1 cm);

pair A, B, O, P;

A = (2,0); B = (0,-3); O = (0,0); P = (O + reflect(A,B)*(O))/2;

draw(interp(A,B,-0.2)--interp(A,B,1.2), blue); draw((-1,0)--(3,0)); draw((0,-4)--(0,1)); draw(O--P); draw(rightanglemark(O,P,B,5));

label("$x$", (3,0), E); label("$y$", (0,1), N); label("$\ell$", interp(A,B,-0.2), N); label("$2$", (O + P)/2, SW, red); [/asy]

Solution

To find the area of the triangle formed by line $\ell$ and the coordinate axes, we need to determine the points where the line intersects the $x$ -axis and $y$ -axis.

Given that the slope of line $\ell$ is $3$ , we know that the equation of the line can be written in slope-intercept form: $y = 3x + b$ Since the line passes through the origin (the intersection of the $x$ -axis and $y$ -axis), the y-intercept $b = 0$ . Thus, the equation of the line is: $y = 3x$

Step 1: Finding the Intercepts

$x$ -intercept: Set $y = 0$ in the equation $y = 3x$ . $0 = 3x \Rightarrow x = 0$ Since the line passes through the origin, the $x$ -intercept is at $(0, 0)$ .
$y$ -intercept: Set $x = 0$ in the equation $y = 3x$ . $y = 3 \cdot 0 = 0$ The $y$ -intercept is also at $(0, 0)$ .

However, this doesn’t yet give us the specific intersection points needed for a bounded triangle with the axes. Given that the problem statement refers to a line with a slope of $3$ and implies a triangle formation, let's assume that the line reaches specific intercepts on the axes based on a typical triangle setup with finite dimensions.

From the diagram (if it depicts points at particular values, like $x = 2$ ), additional calculations may be necessary based on these explicit coordinates

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Coordinate Geometry
Slope
Area of Triangle

Formulas

Slope-intercept form of a line: $ y = mx + b $
Area of a triangle with vertices on axes: $ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $

Theorems

Slope-intercept form theorem
Area of triangle in coordinate geometry

Suitable Grade Level

Grades 8-10

Related Recommendation

Area of Triangle Formed by Line and Coordinate Axes with Slope 3

Find the Area of a Triangle Bounded by Two Lines and the X-Axis

Find the Area of the Green Region in Square ABCD Using Coordinate Geometry

Calculate the Area and Perimeter of a Triangle Formed by Axes and a Line

Calculate Area of Triangle AOB Formed by Line through Points P(-2, 3) and Q(8, 5)