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.



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). △ABC≅△CEA so area of △ABC= area of △CEA. Therefore area of △ABC=12 area of parallelogram ABCE=12bh.
Figure 7.3.2: Draw AE and CE so that ABCE is a parallelogram.
Find the area:
Solution
A=12bh=12(9)(4)=12(36)=18.
Answer: 18.
Find the area to the nearest tenth:
Solution
Draw the height h as shown in Figure 7.3.3

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
Find the area and perimeter:
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=√9√5=3√5
Perimeter = AB+AC+BC=10+5+3√5=15+3√5
Answer: A=15,P=15+3√5.
Find the area and perimeter:
Solution
∠A=∠B=30∘ so △ABC is isosceles with BC=AC=10. Draw height h as in Figure 7.3.4.

△ACD is a 30∘−60∘−90∘ triangle hence
hypotenuse=2(short leg)10=2h5=hlong leg=(short leg)(√3)AD=h√3=5√3.
Similarly BD=5√3.
Area = 12bh=12(5√3+5√3)(5)=12(10√3)(5)=12(50√3)=25√3.
Perimeter = 10+10+5√3+5√3=20+10√3.
Answer: A=25√3,P=20+10√3.
Problems
1 - 4. Find the area of △ABC:
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5 - 6. Find the area to the nearest tenth:
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7 - 20. Find the area and perimeter of △ABC:
7.
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10.
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12.
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15.
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19 - 20. Find the area and perimeter to the nearest tenth:
19.
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21. Find x if the area of △ABC is 35:
22. Find x if the area of △ABC is 24.
23. Find x if the area of △ABC is 12:
24. Find x if the area of △ABC is 108: