Answer:
4) 16√3 in²
5) 63 cm²
Step-by-step explanation:
The formula to use in these cases is ...
A = (1/2)ab·sin(θ)
where a, b are the side lengths and θ is the angle between them.
It helps to know the trig functions of the "special" angles used here.
sin(120°) = sin(60°) = (√3)/2
cos(60°) = 1/2
sin(135°) = cos(45°) = (√2)/2
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4) The external angle at the base is the supplement of 120°, so is 60°. Then the length of the missing segment between the end of the base and the right angle at h is ...
x = (8 in)cos(60°) = (8 in)(1/2) = 4 in
So, the bottom edge of the triangle is 12 in - 4 in = 8 in.
The area is ...
A = (1/2)(8 in)(8 in)sin(120°) = (1/2)64(√3)/2 in² = 16√3 in²
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5) As in the previous problem, the difference between the given horizontal dimension and the base of the triangle is ...
x = (18 cm)cos(180°-135°) = 18(√2)/2 cm = 9√2 cm
Then the base of the triangle is ...
16√2 cm -9√2 cm = 7√2 cm
The area is then ...
A = (1/2)(18 cm)(7√2 cm)(√2)/2 = 63 cm²
