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Answers:
- Problem 7) 25.2 feet
- Problem 8) Angle = 52 degrees; Distance = 14.3 feet
- Problem 9) DE = 22, EC = 10, BC = 20
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Explanation:
The diagrams for problems 7 and 8 are shown below.
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Problem 7
Apply the tangent ratio to find that...
tan(angle) = opposite/adjacent
tan(42) = x/28
28*tan(42) = x
x = 28*tan(42)
x = 25.2113132403396
x = 25.2
Make sure your calculator is in degree mode. One way to check is to type in tan(45) and you should get 1 as a result.
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Problem 8, part 1
To find the angle y, we must use the sine ratio
sin(angle) = opposite/hypotenuse
sin(y) = 18/23
y = arcsin(18/23)
y = 51.5000495907521
y = 52 degrees approximately
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Problem 8, part 2
We can use the pythagorean theorem to find the missing side x
a^2+b^2 = c^2
18^2+x^2 = 23^2
324+x^2 = 529
x^2 = 529-324
x^2 = 205
x = sqrt(205)
x = 14.3178210632763
x = 14.3
Or alternatively, we can apply the tangent ratio on the angle y we found earlier to help find x
tan(angle) = opposite/adjacent
tan(y) = 18/x
tan(51.50004959) = 18/x
x*tan(51.50004959) = 18
x = 18/tan(51.50004959)
x = 14.3178210636621
x = 14.3
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Problem 9
The double tickmarks show that BE = EC = 10. That means BC = BE+EC = 10+10 = 20.
Since we have similar triangles, we can solve for x like so
DE/AC = BE/BC
x/44 = 10/20
20x = 44*10
20x = 440
x = 440/20
x = 22
So DE is 22 units long. Note how this is half as long as the side AC = 44.
