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A 10-foot ladder leans against a vertical wall and forms a 6
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26 Sep 2018, 15:43
26 Sep 2018, 15:43
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Question Stats:
74% (00:50) correct
25% (00:27) wrong based on 54 sessions
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A 10-foot ladder leans against a vertical wall and forms a 60-degree angle with the floor. If the ground below the ladder is horizontal, how far above the ground is the top of the ladder?
(A) \(5\) feet
(B) \(5\sqrt{3}\) feet
(C) \(7.5\) feet
(D) \(10\) feet
(E) \(10\sqrt{3}\) feet
(A) \(5\) feet
(B) \(5\sqrt{3}\) feet
(C) \(7.5\) feet
(D) \(10\) feet
(E) \(10\sqrt{3}\) feet
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Re: A 10-foot ladder leans against a vertical wall and forms a 6
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13 Apr 2020, 15:15
13 Apr 2020, 15:15
1
Expert Reply
For geometry word problems, always reduce the problem to basic shapes when you can, and identify what you have to find. Drawing can help too! In this problem, the figure is a right triangle, and you're asked to find the height, given a hypotenuse of 10.
Next, look for clues in the question and the answer choices.
Because there's a 60° in the question, and √3 shows up in two answer choices, you know that you're being tested on Special Right Triangles: specifically the 30-60-90 triangle.
To solve:
1) Know your ratios! For a 30-60-90 triangle, sides are in the ratio of x : x√3 : 2x for the sides opposite the respective angles. One quick way of thinking about it is: the short side is half the hypotenuse, and the middle side (the height, in this question) is the short side x √3
2) Since the hypotenuse is 10, the short side (the distance from the base of the ladder to the wall) is half of 10, which is 5, so the height is 5 x √3 = 5 √3
Answer: B
Next, look for clues in the question and the answer choices.
Because there's a 60° in the question, and √3 shows up in two answer choices, you know that you're being tested on Special Right Triangles: specifically the 30-60-90 triangle.
To solve:
1) Know your ratios! For a 30-60-90 triangle, sides are in the ratio of x : x√3 : 2x for the sides opposite the respective angles. One quick way of thinking about it is: the short side is half the hypotenuse, and the middle side (the height, in this question) is the short side x √3
2) Since the hypotenuse is 10, the short side (the distance from the base of the ladder to the wall) is half of 10, which is 5, so the height is 5 x √3 = 5 √3
Answer: B
Re: A 10-foot ladder leans against a vertical wall and forms a 6
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11 May 2020, 06:11
11 May 2020, 06:11
A 10-foot ladder leans against a vertical wall and forms a 60-degree angle with the floor. If the ground below the ladder is horizontal, how far above the ground is the top of the ladder?
Clearly, the ladder is forming a 30-60-90 triangle with the wall, the angle between the wall and the floor being 90 degs, and we know that the angle between the bottom of the ladder and the floor is 60 degs. Therefore the angle between the top of the ladder and the wall is 30 degs. The ladder is the hypotenuse of this triangle and we know that it is 10 feet long. The lengths of the hypotenuse, the longer leg and the shorter leg in a 30-60-90 triangle such as this one are in a ratio of 2x, x.root(3) and x.
The hypotenuse is 10 feet.
The shorter leg is 5 feet.
The longer leg is 5.root(3)
The longer leg is the answer to the question. Answer is B.
Clearly, the ladder is forming a 30-60-90 triangle with the wall, the angle between the wall and the floor being 90 degs, and we know that the angle between the bottom of the ladder and the floor is 60 degs. Therefore the angle between the top of the ladder and the wall is 30 degs. The ladder is the hypotenuse of this triangle and we know that it is 10 feet long. The lengths of the hypotenuse, the longer leg and the shorter leg in a 30-60-90 triangle such as this one are in a ratio of 2x, x.root(3) and x.
The hypotenuse is 10 feet.
The shorter leg is 5 feet.
The longer leg is 5.root(3)
The longer leg is the answer to the question. Answer is B.
