Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction 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.

Published byTerence Bell Modified over 8 years ago

Similar presentations

Presentation on theme: "Introduction 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."— Presentation transcript:

1 Introduction 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. 1 6.2.2: Calculating Perimeter and Area

2 Key Concepts 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.). 2 6.2.2: Calculating Perimeter and Area

3 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. 3 6.2.2: Calculating Perimeter and Area Calculating the Perimeter of a Polygon 1.Calculate the length of each side of the polygon using the distance formula:. 2.Add all the lengths of the polygon to find the perimeter. 3.Simplify and approximate the value if necessary. Be sure to include the appropriate label in your answer.

4 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. 4 6.2.2: Calculating Perimeter and Area

5 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 (units 2, feet 2, inches 2, meters 2, centimeters 2, etc.). 5 6.2.2: Calculating Perimeter and Area

6 Key Concepts, continued 6 6.2.2: Calculating Perimeter and Area Calculating the Area of a Triangle 1.Find the equation of the line that represents the base of the triangle. 2.Find the equation of the line that represents the height of the triangle. 3.Find the point of intersection of the line representing the height of the triangle and the line representing the base of the triangle. 4.Calculate the length of the base of the triangle using the distance formula:. 5.Calculate the length of the height of the triangle using the distance formula:. 6.Calculate the area using the formula. Be sure to include the appropriate label in your answer.

7 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 (units 2, feet 2, inches 2, meters 2, centimeters 2, etc.). 7 6.2.2: Calculating Perimeter and Area

8 Calculating the Area of a Rectangle 1.Calculate the length of the base of the rectangle using the distance formula:. 2.Calculate the length of the height of the rectangle using the distance formula:. 3.Calculate the area using the formula. Key Concepts, continued 8 6.2.2: Calculating Perimeter and Area

9 Common Errors/Misconceptions 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 9 6.2.2: Calculating Perimeter and Area

10 Guided Practice 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. 10 6.2.2: Calculating Perimeter and Area

11 Guided Practice: Example 2, continued 1.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. 11 6.2.2: Calculating Perimeter and Area

12 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. 12 6.2.2: Calculating Perimeter and Area

13 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. 13 6.2.2: Calculating Perimeter and Area

14 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. 14 6.2.2: Calculating Perimeter and Area

15 Guided Practice: Example 2, continued 2.Calculate the perimeter of quadrilateral ABCD. Find the sum of the sides of the quadrilateral. perimeter = AB + BC + CD + DA The perimeter of quadrilateral ABCD is units. 15 6.2.2: Calculating Perimeter and Area ✔

16 Guided Practice: Example 2, continued 16 6.2.2: Calculating Perimeter and Area

17 Guided Practice Example 3 Triangle ABC has vertices A (1, –1), B (4, 3), and C (5, –3). Calculate the area of triangle ABC. 17 6.2.2: Calculating Perimeter and Area

18 Guided Practice: Example 3, continued 1.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. 18 6.2.2: Calculating Perimeter and Area

19 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. 19 6.2.2: Calculating Perimeter and Area

20 Guided Practice: Example 3, continued Write the equation of the line that represents side. y – y 1 = m(x – x 1 )Point-slope formula Substitute for m. Substitute (1, –1) for (x 1, y 1 ) Simplify. 20 6.2.2: Calculating Perimeter and Area

21 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. 21 6.2.2: Calculating Perimeter and Area

22 Guided Practice: Example 3, continued 2.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. 22 6.2.2: Calculating Perimeter and Area

23 Guided Practice: Example 3, continued y – y 1 = m(x – x 1 )Point-slope form y – y 1 = 2(x – x 1 )Substitute 2 for m. y – (3) = 2(x – (4))Substitute (4, 3) for (x 1, y 1 ). y – 3 = 2(x – 4)Simplify. y – 3 = 2x – 8Distribute 2 over (x – 4). y = 2x – 5Add 3 to both sides. The equation of the line that represents the height of the triangle is y = 2x – 5. 23 6.2.2: Calculating Perimeter and Area

24 Guided Practice: Example 3, continued 3.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. 24 6.2.2: Calculating Perimeter and Area

25 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. 25 6.2.2: Calculating Perimeter and Area

26 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. 26 6.2.2: Calculating Perimeter and Area

27 Guided Practice: Example 3, continued The point of intersection has a y-value of. The point of intersection is. 27 6.2.2: Calculating Perimeter and Area

28 Guided Practice: Example 3, continued 4.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. 28 6.2.2: Calculating Perimeter and Area

29 Guided Practice: Example 3, continued 5.Calculate the length of the height from point B to the point of intersection. Distance formula Substitute (4, 3) and. Simplify as needed. 29 6.2.2: Calculating Perimeter and Area

30 Guided Practice: Example 3, continued The length of the height is units. 30 6.2.2: Calculating Perimeter and Area

31 Guided Practice: Example 3, continued 6.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. 31 6.2.2: Calculating Perimeter and Area

32 Guided Practice: Example 3, continued = 11 The area of triangle ABC is 11 square units. 32 6.2.2: Calculating Perimeter and Area ✔

33 Guided Practice: Example 3, continued 33 6.2.2: Calculating Perimeter and Area

Download ppt "Introduction 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."

Similar presentations

Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details, and even bicycle frames. Isosceles.

Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details, and even bicycle frames. Isosceles.
Introduction What does it mean to be opposite? What does it mean to be consecutive? Think about a rectangular room. If you put your back against one corner.

Introduction What does it mean to be opposite? What does it mean to be consecutive? Think about a rectangular room. If you put your back against one corner.
Preview Warm Up California Standards Lesson Presentation.

Preview Warm Up California Standards Lesson Presentation.
Introduction It is not uncommon for people to think of geometric figures, such as triangles and quadrilaterals, to be separate from algebra; however, we.

Introduction It is not uncommon for people to think of geometric figures, such as triangles and quadrilaterals, to be separate from algebra; however, we.
Introduction The slopes of parallel lines are always equal, whereas the slopes of perpendicular lines are always opposite reciprocals. It is important.

Introduction The slopes of parallel lines are always equal, whereas the slopes of perpendicular lines are always opposite reciprocals. It is important.
Midpoints of line segments. Key concepts  Line continue infinitely in both directions, their length cannot be measured.  A Line Segment is a part of.

Midpoints of line segments. Key concepts  Line continue infinitely in both directions, their length cannot be measured.  A Line Segment is a part of.
Introduction 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.

Introduction 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.
2.5Formulas and Additional Applications from Geometry

2.5Formulas and Additional Applications from Geometry
Using the Pythagorean Theorem

Using the Pythagorean Theorem
Introduction Previously, we learned how to solve quadratic-linear systems by graphing and identifying points of intersection. In this lesson, we will focus.

Introduction Previously, we learned how to solve quadratic-linear systems by graphing and identifying points of intersection. In this lesson, we will focus.
Polygons, Circles, and Solids

Polygons, Circles, and Solids
Lesson 6.3 – Finding Perimeter and Area in the Coordinate Plane Concept: Distance in the Coordinate Plane EQ: how do we find area & perimeter in the.

Lesson 6.3 – Finding Perimeter and Area in the Coordinate Plane Concept: Distance in the Coordinate Plane EQ: how do we find area & perimeter in the.
Definitions and formulas for the shapes you love Perimeter and Area.

Definitions and formulas for the shapes you love Perimeter and Area.
Section 1-7 Perimeter, Area, Circumference SPI 21B: solve equations to find length, width, perimeter and area SPI 32L: determine the area of indicated.

Section 1-7 Perimeter, Area, Circumference SPI 21B: solve equations to find length, width, perimeter and area SPI 32L: determine the area of indicated.
You will learn to solve problems that involve the perimeters and areas of rectangles and parallelograms.

You will learn to solve problems that involve the perimeters and areas of rectangles and parallelograms.
Divide each side by 2. Write original equation. Write 3x + 2y = 8 so that y is a function of x. EXAMPLE 2 Rewrite an equation Subtract 3x from each side.

Divide each side by 2. Write original equation. Write 3x + 2y = 8 so that y is a function of x. EXAMPLE 2 Rewrite an equation Subtract 3x from each side.
Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.

Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.
Introduction Literal equations are equations that involve two or more variables. Sometimes it is useful to rearrange or solve literal equations for a specific.

Introduction Literal equations are equations that involve two or more variables. Sometimes it is useful to rearrange or solve literal equations for a specific.
Perimeter Is the sum of the lengths of the sides. When solving a perimeter problem, it is helpful to draw and label a figure to model the region.

Perimeter Is the sum of the lengths of the sides. When solving a perimeter problem, it is helpful to draw and label a figure to model the region.
Linear Measure and Precision

Linear Measure and Precision

Similar presentations

Ads by Google