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7.3: The Area of a Triangle

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For each of the triangles in Figure 7.3.1, side AB is called the base and CD is called the height or altitude drawn to this base. The base can be any state of the triangle though it is usually chosen to be the side on which the triangle appears to be resting. The height is the line drawn perpendicular to the base from the opposite vertex. Note that the height may fall outside the triangle, If the triangle is obtuse, and that the height may be one of the legs, if the triangle is a right triangle.

clipboard_efd670e805c28238bdc4efc37dd102001.png clipboard_e6101f70eb26d7d2e6dd656f6e0849fc5.png clipboard_e7539c3e97e505b76ecea452531f7aa9b.png
Figure 7.3.1: Triangles with base b and height h.
Theorem 7.3.1

The area of a triangle is equal to one-half of its base times its height.

A=12bh

Proof

For each of the triangles illustrated in Figure 7.3.1, draw AE and CE so that ABCE is a parallelogram (Figure PageIndex2). ABCCEA so area of ABC= area of CEA. Therefore area of ABC=12 area of parallelogram ABCE=12bh.

Screen Shot 2020-12-18 at 4.53.29 PM.pngFigure 7.3.2: Draw AE and CE so that ABCE is a parallelogram.
Example 7.3.1

Find the area:

Screen Shot 2020-12-18 at 4.54.27 PM.png

Solution

A=12bh=12(9)(4)=12(36)=18.

Answer: 18.

Example 7.3.2

Find the area to the nearest tenth:

Screen Shot 2020-12-18 at 4.55.58 PM.png

Solution

Draw the height h as shown in Figure 7.3.3

Screen Shot 2020-12-18 at 4.56.39 PM.pngFigure 7.3.3: Draw height h.

sin40=h10.6428=h10(10)(.6428)=h10(10)6.428=h

Area = 12bh=12(15)(6.428)=12(96.420)=48.21=48.2

Answer: A=48.2

Example 7.3.3

Find the area and perimeter:

Screen Shot 2020-12-18 at 5.00.27 PM.png

Solution

A=12bh=12(5)(6)=12(30)=15.

To find AB and BC we use the Pythagorean theorem:

AD2+BD2=AB282+62=AB264+36=AB2100=AB210=AB CD2+BD2=BC232+62=BC29+36=BC245=BC2BC=45=95=35

Perimeter = AB+AC+BC=10+5+35=15+35

Answer: A=15,P=15+35.

Example 7.3.4

Find the area and perimeter:

Screen Shot 2020-12-18 at 5.08.58 PM.png

Solution

A=B=30 so ABC is isosceles with BC=AC=10. Draw height h as in Figure 7.3.4.

Screen Shot 2020-12-18 at 5.11.27 PM.pngFigure 7.3.4: Draw height h.

ACD is a 306090 triangle hence

hypotenuse=2(short leg)10=2h5=hlong leg=(short leg)(3)AD=h3=53.

Similarly BD=53.

Area = 12bh=12(53+53)(5)=12(103)(5)=12(503)=253.

Perimeter = 10+10+53+53=20+103.

Answer: A=253,P=20+103.

Problems

1 - 4. Find the area of ABC:

1.

Screen Shot 2020-12-18 at 5.22.52 PM.png

2.

Screen Shot 2020-12-18 at 5.23.11 PM.png

3.

Screen Shot 2020-12-18 at 5.23.48 PM.png

4.

Screen Shot 2020-12-18 at 5.24.04 PM.png

5 - 6. Find the area to the nearest tenth:

5.

Screen Shot 2020-12-18 at 5.24.20 PM.png

6.

Screen Shot 2020-12-18 at 5.26.18 PM.png

7 - 20. Find the area and perimeter of ABC:

7.

Screen Shot 2020-12-18 at 5.26.39 PM.png

8.

Screen Shot 2020-12-18 at 5.27.38 PM.png

9.

Screen Shot 2020-12-18 at 5.27.54 PM.png

10.

Screen Shot 2020-12-18 at 5.28.24 PM.png

11.

Screen Shot 2020-12-18 at 5.29.05 PM.png

12.

Screen Shot 2020-12-18 at 5.30.39 PM.png

13.

Screen Shot 2020-12-18 at 5.30.57 PM.png

14.

Screen Shot 2020-12-18 at 5.31.13 PM.png

15.

Screen Shot 2020-12-18 at 5.31.51 PM.png

16.

Screen Shot 2020-12-18 at 5.31.36 PM.png

17.

Screen Shot 2020-12-18 at 5.32.06 PM.png

18.

Screen Shot 2020-12-18 at 5.32.48 PM.png

19 - 20. Find the area and perimeter to the nearest tenth:

19.

Screen Shot 2020-12-18 at 5.33.35 PM.png

20.

Screen Shot 2020-12-18 at 5.34.09 PM.png

21. Find x if the area of ABC is 35:

Screen Shot 2020-12-18 at 5.36.02 PM.png

22. Find x if the area of ABC is 24.

Screen Shot 2020-12-18 at 5.36.23 PM.png

23. Find x if the area of ABC is 12:

Screen Shot 2020-12-18 at 5.36.50 PM.png

24. Find x if the area of ABC is 108:

Screen Shot 2020-12-18 at 5.37.17 PM.png


This page titled 7.3: The Area of a Triangle is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Henry Africk (New York City College of Technology at CUNY Academic Works) .

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