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Applications of
Algebraic Modeling 2
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2. Copyright © Cengage Learning. All rights reserved.
Section
2-2
Models and Patterns
in Triangles
P.1
Velocity
Copyright © Cengage Learning. All rights reserved.

3. Models and Patterns in Triangles
A triangle is the simplest of the polygons, having only three sides and three angles. Angles are measured in degrees and classified according to the number of degrees in each.
An acute angle measures between 0° and 90°; a right angle measures exactly 90°; an obtuse angle measures between 90° and 180°. Similarly, a triangle can be classified by the sizes of its angles.
While the sum of the three angles in every triangle is 180°, if all angles are acute angles, the triangle is classified as an acute triangle.

4. Models and Patterns in Triangles
If one of the angles is an obtuse angle, the triangle is classified as an obtuse triangle.
If one of the angles is a 90° angle or right angle, the triangle is classified as a right triangle.
These classifications of triangles are illustrated in Figure 2-7.
Figure 2-7

5. Models and Patterns in Triangles
Triangles can also be classified by the lengths of their sides. A triangle having three equal sides is called an equilateral triangle. In addition to having three equal sides, an equilateral triangle has three equal angles each measuring 60°.
An isosceles triangle has two sides that are the same length and two angles that are the same size.
A scalene triangle has three sides that are all different lengths and three angles of different measurements.

6. Models and Patterns in Triangles
These classifications of triangles are illustrated in Figure 2-8.
Figure 2-8

7. Models and Patterns in Triangles
Each corner of a triangle has a point called the vertex (plural: vertices). Every triangle has three vertices.
The base of a triangle can be any one of the three sides but is usually the one drawn at the bottom of the triangle. In an isosceles triangle, the base is usually the unequal side.

8. Models and Patterns in Triangles
The height of a triangle is a perpendicular line segment from the base to the opposite vertex. Look at the labeled parts of  ABC shown in Figure 2-9.
Figure 2-9

9. Models and Patterns in Triangles
The area of any flat surface is the amount of surface enclosed by the sides of the figure.
The units associated with area are square units such as cm2, ft2, or m2. In this section, we will look at two area formulas that can be used to calculate the area of a triangle.

10. Models and Patterns in Triangles
In order to calculate the area of a triangle using this formula, the lengths of the base b and the height h must be known.

11. Example 2 – Calculating the Area of a Triangle
Use the area formula A = 0.5bh, where b = base and h = height, to calculate the area of the given triangles.
(a)
Since this triangle is an isosceles triangle, the height divides the base into two equal parts.

12. Example 2 – Calculating the Area of a Triangle
cont’d
Before we use the formula, we must calculate the length of the base.
Base = 2(6 in.) = 12 in.
h = 8 in. b = 12 in.
A = 0.5(12 in.)(8 in.) = 48 in.2

13. Example 2 – Calculating the Area of a Triangle
cont’d
(b)
In a right triangle, the legs serve as the base and height of the triangle.
h = 9 cm b = 6 cm
A = 0.5(6 cm)(9 cm) = 27 cm2

14. Example 2 – Calculating the Area of a Triangle
cont’d
(c)
h = 6 in. b = 10 in.
A = 0.5(10 in.)(6 in.) = 30 in.2
In many real-life applications where we are required to find the area of an object in the shape of a triangle, we may not be able to find the height of the triangle.

15. Example 2 – Calculating the Area of a Triangle
cont’d
Another formula, called Hero’s formula, can also be used to find the area of a triangle using only the lengths of the sides of the triangle.

16. Models and Patterns in Triangles

17. Example 5 – Area of the Wall of a Shed
Calculate the area of the side of the shed, including the door and window, as shown in Figure 2-12.
We will subdivide the figure into two parts, a rectangle and a triangle, calculate the areas of each, and total the areas.
Figure 2-12

18. Example 5 – Area of the Wall of a Shed
cont’d
Because we are not given a height for the triangular portion, we must use Hero’s formula to calculate the area.
(a) Calculate the areas of the rectangle having l = 15 feet and w = 10.8 feet.
A = lw = (15 ft)(10.8 ft)
= 162 ft2
(b) Calculate the area of the triangle using Hero’s formula Let a = 9.5 feet, b = 9.5 feet, and c =15 feet.

19. Example 5 – Area of the Wall of a Shed
cont’d
Round the answer to the nearest tenth.
Step 1

20. Example 5 – Area of the Wall of a Shed
cont’d
Step 2

21. Example 5 – Area of the Wall of a Shed
cont’d
(c) Find the total area.