Problem with the triangle PQT (altitude problem)
Part A
Answer is 60 degrees.
You can solve x+30+90 = 180 or you can solve x+30 = 90 to find that x = 60. Note how angle PTR and angle TPR are complementary angles. A shortcut is to do 90-30 = 60.
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Part B
The exact length of PR is 7 inches
Triangle TPR is a 30-60-90 triangle, so the short leg is half that of the hypotenuse (14). Take half of 14 to get 7. The short leg is always opposite the 30 degree angle for a 30-60-90 triangle.
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Part C
Answer: 7*sqrt(3)
The notation "sqrt" is shorthand for "square root"
TR is the long leg of the right triangle TPR. The long leg of a 30-60-90 triangle is equal to sqrt(3) times the short leg. Alternatively, you can use the pythagorean theorem to solve for the long leg.
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Part D
Answer: 7*sqrt(2)
Triangle PRQ is a right triangle. It is a 45-45-90 right triangle. The hypotenuse is equal to sqrt(2) times the leg 7. Alternatively, you can use the pythagorean theorem to solve for the hypotenuse.
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Problem with the tree and support wire
Part A
Answer: 36.9
Work Shown:
cos(x) = adjacent/hypotenuse
cos(x) = 24/30
x = arccos(24/30)
x = 36.8698976
x = 36.9
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Part B
Answer: 18
Use the pythagorean theorem to find that
a^2 + b^2 = c^2
a^2 + 24^2 = 30^2
a^2 + 576 = 900
a^2 = 900-576
a^2 = 324
a = sqrt(324)
a = 18