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Chapter 13: Problem 4

Find the area of the square with vertices at \((1,2),(6,2),(6,7)\) and \((1,7)\)

Short Answer

Expert verified
The area of the square is 25 square units.

Step by step solution

01

Identify the Length of a Side

Start by finding the length of one side of the square. This can be done by choosing two points that lie on the same axis (either 'x' or 'y'). In this case, the points \((1,2)\) and \((6,2)\) lie on the same 'y' axis. The length can be calculated by subtracting the 'x' coordinates of these two points, \(6 - 1 = 5\). Thus, the length of one side of the square is 5 units.
02

Calculate Area of the square

Once the length of a side is found, the area of a square can be calculated using the formula \( \text{Area} = \text{side}^{2} \). Substituting the length of the side into the equation gives: \( \text{Area} = 5^{2} = 25 \) square units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square area
When it comes to finding the area of a square, you're simply looking at the amount of space contained within its boundaries. Imagine you're trying to figure out how many tiles it would take to completely fill the square's surface.
The area is always expressed in square units, such as square meters or square centimeters.
Squares are unique in that all their sides are equal, which simplifies how we calculate the area.
  • First, determine the length of one of its sides.
  • Remember, since all sides of a square are the same, you only need to measure one.
  • Finally, square this number to find the total area.
This concept is what makes squares so straightforward when comparing them to other polygons like rectangles or triangles.
coordinate geometry
Coordinate geometry, also known as analytic geometry, melds algebra and geometry. It allows us to use coordinates to represent and explore geometric figures within the Cartesian plane.
This is incredibly useful for solving problems involving shapes and their properties, because it lets us use mathematical formulas directly on coordinate points.
For example, if you are tasked with finding the dimensions of a shape with specific vertices, like a square, coordinate geometry will serve you well.
  • This involves identifying coordinates, such as (1,2), (6,2), (6,7), and (1,7).
  • You can calculate the distances between these points.
  • In our problem, notice how coordinates share the same 'y' or 'x' value.
This is a crucial clue that two points lie on the same axis, a detail pivotal for finding line lengths.
formula for area of square
The formula for finding the area of a square is one of the simplest in geometry.
It's central to many practical applications and often one of the first formulas students learn in geometry because of its ease and usefulness.
Here鈥檚 a quick breakdown of how it works:
  • To find the area, take the length of one side of the square.
  • Square this number鈥攎eaning, multiply the length by itself.
Mathematically, this is expressed as \( \text{Area} = \text{side}^{2} \).
Applying this formula, from the given points, the side length of the square is 5, leading to \( 5^2 = 25 \) square units as the area.

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Most popular questions from this chapter

Determine the intersection of the solution sets of the two inequalities \(y>2\) and \(x+2 y<6\) by graphing.

Consider the points \(A=(2,3,-5), B=(8,9,1),\) and \(\mathrm{C}=(3,17,1)\) a Find the midpoint of \(\overline{\mathrm{AB}}\). b Find, to the nearest tenth, the length of the median from C to \(\overline{\mathrm{AB}}\).

Graph each equation. \(\mathbf{a.} \times^{2}+y^{2}=9\) b. \((x-1)^{2}+(y+2)^{2}=16\)

The graph of \(x^{2}+y^{2}=25\) is a circle. (Circular graphs will be studied later in this chapter.) The graph of \(x^{2}-y^{2}=7\) is a hyperbola. (Hyperbolas are normally studied in a later math course.) Use one of the methods of solving a system of equations to find the intersection of the circle and the hyperbola.

If \(\mathrm{H}=(10,2)\) and \(\mathrm{K}=(18,17)\) and if \(\mathrm{Jis}\) any point on the graph of \(x=2,\) find, to the nearest tenth, the minimum distance from H to J to K.
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