Question

Write a system of inequalities whose graphed solution set is an isosceles triangle.

Short answer

The system of inequalities that forms an isosceles triangle when graphed is: y ≥ 0, y ≤ -2x + 2, and y ≤ 2x.

Step-by-step solution

Identifying the isosceles triangle

First, assign locations for your isosceles triangle on a standard graph. A suggestion is to have vertices of the triangle at points (0, 0), (2, 0) and (1, 2). This forms an isosceles triangle where the base is on the x-axis between points (0,0) and (2,0), and the equal sides run from each of these points up to (1,2).

Formulate the inequalities

The three lines of the triangle will be represented by the equations y ≥ 0 (which is the x-axis), y ≤ -2x + 2 (the line that goes through the points (1,2) and (0,0)), and y ≤ 2x (the line that goes through the points (1, 2) and (2,0)). Make sure that the signs are set so that the feasible set of points determined by the inequalities is the triangle itself, and not all points outside of the triangle.

Final step

Now the system of inequalities that, when graphed, result in an isosceles triangle has been obtained, and it is: y ≥ 0, y ≤ -2x + 2, and y ≤ 2x.

Key concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle where at least two sides are of equal length. This creates a symmetry that is not present in scalene triangles, which have all sides of different lengths. In the context of coordinate geometry, this means that the coordinates of the triangle's vertices (corners) must satisfy certain conditions to ensure that two sides are equal. For example, if we're plotting an isosceles triangle on a graph, you might place vertices at points that help to visually confirm that two sides are indeed the same length. In our exercise, we place the vertices at (0, 0), (2, 0), and (1, 2) to form an isosceles triangle.

• The base is horizontal from (0, 0) to (2, 0).
• The two equal sides are from (0, 0) to (1, 2) and (2, 0) to (1, 2).

The equal sides create an aesthetic and mathematical symmetry that can be represented algebraically, making it essential for graphing in systems of inequalities.

System of Inequalities

Graphing a system of inequalities involves finding a set of solutions that simultaneously satisfy all given conditions. Inequalities are different from equations as they do not represent precise line intersections, but rather areas where a solution is valid. To construct an isosceles triangle using inequalities, you must identify the conditions defining each line segment part of the triangle and how they envelop the area within.

• The inequality \( y \geq 0 \) represents the area above the x-axis.
• The inequality \( y \leq -2x + 2 \) corresponds to the line forming one side of the triangle.
• Finally, \( y \leq 2x \) covers the opposite side.

By combining these inequalities, we restrict the feasible region to only include points inside the triangle, effectively using boundaries to outline the shape.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, utilizes a coordinate system to handle geometric problems algebraically. It allows us to express geometric figures, like triangles, as algebraic equations or inequalities. For instance, when plotting an isosceles triangle, we leverage the x and y axes to assign precise locations to each vertex.

• The coordinate (0, 0) often serves as a starting or reference point.
• Using points like (2, 0) ensures alignment along a base axis, here the x-axis.
• Finally, a vertex such as (1, 2) provides height and symmetry.

This enables not only visualization but also calculation of important metrics like area and side length through algebraic means, crucial for solving complex problems involving systems of inequalities. With the triangle's sides represented by inequalities, coordinate geometry elegantly ties together visual and algebraic understanding.