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The distance formula can be used to find solutions to many real-world problems. In the previous lesson, the distance formula was used to find the distance between two given points. In this lesson, the distance formula will be applied to perimeter and area problems.

A polygon is a two-dimensional figure formed by three or more segments. Sometimes we need to calculate the perimeter or distance around a polygon, as well as find the area or the amount of space inside the boundary of a polygon. The distance formula is a valuable tool for both of these calculations.

6.2.2: Calculating Perimeter and Area

Situations where you would need to calculate perimeter include finding the amount of linear feet needed to fence a yard or a garden, determining the amount of trim needed for a room, or finding the amount of concrete needed to edge a statue.

Perimeter is the sum of the lengths of all the sides of a polygon.

The final answer must include the appropriate label (units, feet, inches, meters, centimeters, etc.).

6.2.2: Calculating Perimeter and Area

Key Concepts, continued

Sometimes the answer is not a whole number. If it is not, you must simplify the radical and then approximate the value.

6.2.2: Calculating Perimeter and Area

Key Concepts, continued

Calculating area is necessary when finding the amount of carpeting needed for a room in your home, or to determine how large a garden will be.

The area of a triangle is found using the formula .

The height of a triangle is the perpendicular distance from the third vertex to the base of the triangle.

6.2.2: Calculating Perimeter and Area

Key Concepts, continued

It may be necessary to determine the equation of the line that represents the height of the triangle before calculating the area. For an example of this, see Example 3 in the Guided Practice.

Determining the lengths of the base and the height is necessary if these lengths are not stated in the problem.

The final answer must include the appropriate label (units2, feet2, inches2, meters2, centimeters2, etc.).

6.2.2: Calculating Perimeter and Area

Key Concepts, continued

6.2.2: Calculating Perimeter and Area

Key Concepts, continued

By definition, rectangles have adjacent sides that are perpendicular.

The area of a rectangle is found using the formula Area = (base)(height).

The lengths of the base and height are found using the distance formula.

The final answer must include the appropriate label (units2, feet2, inches2, meters2, centimeters2, etc.).

6.2.2: Calculating Perimeter and Area

Key Concepts, continued

6.2.2: Calculating Perimeter and Area

forgetting to simplify radicals

incorrectly simplifying radicals

adding x-values and y-values rather than subtracting them when using the distance formula

incorrectly finding the height of a triangle

6.2.2: Calculating Perimeter and Area

Example 2

Quadrilateral ABCD has vertices A (–3, 0), B (2, 4), C (3, 1), and D (–4, –3). Calculate the perimeter of the quadrilateral.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 2, continued

Calculate the length of each side of the quadrilateral using the distance formula.

Calculate the length of .

Distance formula

Substitute (–3, 0) and (2, 4).

Simplify as needed.

The length of is units.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 2, continued

Calculate the length of .

Distance formula

Substitute (2, 4) and (3, 1).

Simplify as needed.

The length of is units.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 2, continued

Calculate the length of .

Distance formula

Substitute (3, 1) and (–4, –3).

Simplify as needed.

The length of is units.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 2, continued

Calculate the length of .

Distance formula

Substitute (–4, –3) and (–3, 0).

Simplify as needed.

The length of is units.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 2, continued

Calculate the perimeter of quadrilateral ABCD.

Find the sum of the sides of the quadrilateral.

perimeter = AB + BC + CD + DA

The perimeter ofquadrilateralABCD is units.

✔

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 2, continued

6.2.2: Calculating Perimeter and Area

Example 3

Triangle ABC has vertices A (1, –1),

B (4, 3), and C (5, –3). Calculate the area of triangle ABC.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Find the equation of the line that represents the base of the triangle.

Let be the base.

Calculate the slope of the equation that represents side .

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Slope formula

Substitute (1, –1) and (5, –3).

Simplify as needed.

The slope of the equation the represents side is .

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Write the equation of the line that represents side .

y – y1 = m(x – x1)Point-slope formula

Substitute for m.

Substitute (1, –1) for (x1, y1)

Simplify.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Distribute over (x – 1).

Subtract 1 from both sides.

The equation of the line that represents the base of the triangle is .

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Find the equation of the line that represents the height of the triangle.

The equation of the line that represents the height is perpendicular to the base; therefore, the slope of this line is the opposite reciprocal of the base.

The slope of the line representing the height is 2.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

y – y1 = m(x – x1) Point-slope form

y – y1 = 2(x – x1) Substitute 2 for m.

y – (3) = 2(x – (4)) Substitute (4, 3) for (x1, y1).

y – 3 = 2(x – 4) Simplify.

y – 3 = 2x – 8 Distribute 2 over (x – 4).

y = 2x – 5 Add 3 to both sides.

The equation of the line that represents the height of the triangle is y = 2x – 5.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Find the point of intersection of the line representing the height and the line representing the base of the triangle.

Set the equation of the line representing the base and the equation of the line representing the height equal to each other to determine the point of intersection.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Set the equations equal to each other.

Add to both sides.

Subtract 2x from both sides.

Divide both sides by .

The point of intersection has an x-value of .

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Substitute into either equation to find the y-value.

y= 2x– 5Equation of the line representing height

Substitute for x.

Simplify.

Solve for y.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

The point of intersection has a y-value of .

The point of intersection is .

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Calculate the length of the base, , of the triangle.

Distance formula

Substitute (1, –1) and (5, –3).

Simplify as needed.

The length of is units.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Calculate the length of the height from point B to the point of intersection.

Distance formula

Substitute (4, 3) and .

Simplify as needed.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

The length of the height is units.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

Calculate the area of triangle ABC.

Area formula for triangles

Substitute the lengths of the height and the base of the triangle.

Simplify as needed.

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

= 11 The area of triangle ABC is 11 square units.

✔

6.2.2: Calculating Perimeter and Area

Guided Practice: Example 3, continued

6.2.2: Calculating Perimeter and Area

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