Question

Problem 1) Do the side lengths 5, 7, 9 make a Pythagorean triple? (1 point)

Yes, the sides 5,7,9 make a Pythagorean triple!

No, the sides 5,7,9 do not make a Pythagorean triple

It cannot be proven either way
Problem 2) Suppose we have three side lengths: 18, 24, and x. What value of x would make this a Pythagorean triple? Assume x is the hypotenuse of the triangle.(1 point)

x = 10

x = 26

x = 30

x = 32
Problem 3) A tree is 24 feet tall and casts a shadow that is 10 feet long (along the ground). What is the distance from the top of the tree to the top of its shadow? Apply a Pythagorean triple to solve this problem. Hint: draw a picture to help.(1 point)

x = 26

x = 30

x = 15

x = 2
Problem 4) A ramp is 16 feet long and 12 feet tall. What is the length of the ramp, x?



(1 point)

x = 12

x = 18

x = 15

x = 20
Problem 5) The sides (11,60,61) are a Pythagorean triple, and we can multiply each point by 3 to get (33,180,183). Is this new set still a Pythagorean triple?(1 point)

Yes, (33,180,183) is still a Pythagorean triple

No, (33,180,183) is not a Pythagorean triple anymore

We can't prove either way
Problem 6) Using an inverse trigonometric function, find the value of angle x.



(1 point)

x = 21.5

x = 46.6

x = 76.3

x = 89.4
Problem 7) John built a ramp that is 10 feet tall and 18 feet long on the slanted surface. Using the inverse sine function, find the measure of angle A.



(1 point)

A = 51.2°

A = 67.1°

A = 33.7°

A = 99.4°
Problem 8) Find the measure of angle x using the inverse tangent function.



(1 point)

x = 29.3°

x = 35.2°

x = 78.3°

x = 52.1°
Problem 9) Sarah is standing at point A and is looking upward at point C. Point A is 8 feet away from point B, and point A is 16 feet away from point C. Use the inverse of cosine to find the measure of angle A.



(1 point)

A = 30°

A = 60°

A = 70°

A =100°
Problem 10) Use the inverse of cosine to find the interior measure of angle A. Round your answer to the nearest whole number.



(1 point)

A = 45°

A = 54°

A = 62°

A = 95°
Problem 11) Solve for the measure of the missing angle using an inverse trigonometric function of your choice.



(1 point)

15.4°

22.6°

45.8°

97.5°
Problem 12) Use the inverse tangent function to find the missing angle, B.



(1 point)

B = 51°

B = 24°

B = 98°

B = 103°
Problem 13) Use the inverse of tangent function to find the measure of the missing angle. Round your answer to the tenths digit.



(1 point)

21.8°

35.7°

49.6°

57.1°
Problem 14) Find the measure of angle y using an inverse trigonometric function.



(1 point)

67.9°

11.4°

53.4°

34.9°
Problem 15) Luke is flying a helicopter at 1,350 feet above the ground. While flying, he sees an airplane that is 2,400 horizontal feet in front of him, and an unknown vertical distance above him. He knows that the angle of elevation from his position to the airplane is 54 degrees. How far above the ground is this unknown airplane? Hint: draw a picture to visualize the problem.(1 point)

4,653 feet

3,303 feet

4,714 feet

5,245 feet
Problem 16) Solve for the horizontal distance of the ramp using a trigonometric function.



(1 point)

10 feet

15 feet

25 feet

40 feet
Problem 17) Solve for x using the Law of Sines.



(1 point)

x = 4

x = 5

x = 8

x = 10
Problem 18) Solve for the value of the missing side, x, using the Law of Sines.



(1 point)

x = 19

x = 10

x = 15

x = 25
Problem 19) A nonright triangle has side lengths of a = 8, b = 15, and c = 20. Find the measure of angle C using the Law of Cosines, as written below.



(1 point)

C = 55°

C = 88°

C = 117°

C = 157°
Problem 20) A nonright triangle has side lengths of a = 17, b = 20, and angle ∠C = 19°. Find the measure of side c using the Law of Cosines, as written below.



(1 point)

c = 2.5

c = 6.7

c = 9.8

c = 10.1
Problem 21) Explain in your own words what is meant by the term "Pythagorean Triple." How can you prove that three numbers are a Pythagorean Triple? Can non-integer values make a Pythagorean Triple?(3 points)

Answer 1

Here are the answers I can determine from the given information. Several problems require the diagram or numeric labels from the figure — I list those and ask you to upload the missing images or give the side/angle measures so I can finish them.

Problem 1
- No, the sides 5,7,9 do not make a Pythagorean triple.
(5^2 + 7^2 = 25 + 49 = 74 ≠ 9^2 = 81)

Problem 2
- x = 30.
(hypotenuse = sqrt(18^2 + 24^2) = sqrt(900) = 30)

Problem 3
- x = 26.
(distance = sqrt(24^2 + 10^2) = sqrt(676) = 26)

Problem 4
- x = 20.
(ramp length = sqrt(12^2 + 16^2) = sqrt(400) = 20)

Problem 5
- Yes, (33,180,183) is still a Pythagorean triple.
(Scaling a Pythagorean triple by 3 preserves a^2 + b^2 = c^2)

Problem 6
- I need the figure or the side/angle measures to compute this inverse-trig value. Please upload the diagram or give the sides.

Problem 7
- A = 33.7°.
(sin A = opposite/hypotenuse = 10/18 = 5/9 → A = arcsin(5/9) ≈ 33.7°)

Problem 8
- I need the figure or numeric opposite/adjacent values to compute tan^-1. Please upload the diagram or give the sides.

Problem 9
- A = 60°.
(cos A = adjacent/hypotenuse = AB/AC = 8/16 = 1/2 → A = arccos(1/2) = 60°)

Problem 10
- I need the figure or the side/hypotenuse values for this inverse-cosine problem. Please upload the diagram or give the numbers.

Problem 11
- I need the figure or numeric side/angle values to choose the correct inverse-trig answer. Please upload or provide them.

Problem 12
- I need the figure or the opposite/adjacent values to compute B = arctan(...). Please upload the diagram or give the measurements.

Problem 13
- I need the figure or the opposite/adjacent values. Please upload the diagram or give the numbers.

Problem 14
- I need the figure or numeric values for y. Please upload the diagram or provide the sides/angles.

Problem 15
- 4,653 feet.
(vertical difference from helicopter to plane = 2400·tan(54°) ≈ 3303.31; add helicopter altitude 1350 → ≈ 4653 ft)

Problem 16
- I need the ramp diagram or the given angle/height to compute the horizontal distance. Please upload the figure or give the numbers.

Problem 17
- I need the triangle (angles/sides) shown in the figure to apply the Law of Sines. Please upload the diagram or provide the labeled measures.

Problem 18
- I need the given angle(s) and side(s) from the figure to apply the Law of Sines. Please upload the diagram or give the values.

Problem 19
- C = 117°.
(cos C = (a^2 + b^2 − c^2) / (2ab) = (64 + 225 − 400) / (240) = −111/240 ≈ −0.4625 → C ≈ arccos(−0.4625) ≈ 117°)

Problem 20
- c = 6.7 (approximately).
(c^2 = 17^2 + 20^2 − 2·17·20·cos19° → c ≈ sqrt(45.447) ≈ 6.742 → 6.7)

Problem 21 (short answer)
- A "Pythagorean triple" is a set of three positive numbers (usually integers) (a, b, c) that satisfy a^2 + b^2 = c^2, so they are the side lengths of a right triangle. To prove three numbers form a Pythagorean triple, square the two shorter numbers, add them, and check whether the sum equals the square of the longest number. While the term is usually used for integer triples (like (3,4,5)), any real numbers a, b, c that satisfy a^2 + b^2 = c^2 form a (more general) Pythagorean triple — but if you mean "Pythagorean triple" in the conventional sense, people usually require integer entries.

If you upload the missing diagrams (or give the specific side/angle measures referenced in problems 6, 8, 10–14, 16–18), I will finish those answers.