For Which Value of x Is the Figure a Rectangle?
Introduction
When a geometry problem asks “for which value of x is the figure a rectangle?” it usually provides a shape whose side lengths, angles, or coordinates are expressed in terms of a variable x. The goal is to determine the specific numeric value of x that makes the shape satisfy the defining properties of a rectangle: opposite sides are equal and parallel, all interior angles are right angles (90°), and the diagonals are congruent. This article walks you through the reasoning process, step‑by‑step, so you can solve similar problems confidently and understand why each condition matters.
1. Understanding the Given Figure
Before solving, you need to translate the visual information into algebraic expressions. A typical textbook diagram might look like this:
- A quadrilateral ABCD with vertices labeled in order.
- Side AB = 2x + 5 units
- Side BC = x + 2 units
- Side CD = 3x − 1 units
- Side DA = x + 2 units (the same as BC, indicating a pair of opposite sides are already equal)
- One interior angle, say ∠ABC, is given as 2x°
The figure is not yet a rectangle because the side lengths are not all consistent with a right‑angled shape, and the angle is not necessarily 90°.
2. Applying Rectangle Properties
A rectangle must meet three core conditions:
- Opposite sides are equal and parallel.
- All interior angles are 90°.
- Diagonals are equal in length.
Because the problem supplies side lengths, we can start with condition 1. But in a rectangle, AB = CD and BC = DA. The diagram already shows BC = DA, so we only need to enforce AB = CD.
Next, condition 2 tells us that any interior angle must be a right angle. The given angle ∠ABC = 2x° must therefore equal 90°.
Condition 3 (equal diagonals) is often automatically satisfied once the first two conditions hold for a quadrilateral, but we can verify it later if needed.
3. Setting Up the Equations
3.1. Equality of Opposite Sides
[ \text{AB} = \text{CD} ]
[ 2x + 5 = 3x - 1 ]
3.2. Right‑Angle Condition
[ \angle ABC = 2x = 90° ]
These two equations give us two possible pathways to the solution. In most textbook problems, both conditions must be satisfied simultaneously, which narrows the answer to the single value that works for both.
4. Solving the Equations
4.1. From the Side‑Equality Equation
[ 2x + 5 = 3x -
