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20 Oct 2021, 02:00
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Difficulty:
45%
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66% (01:56) correct
34%
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wrong
based on 158
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A hiker is trying to determine the height of a vertical cliff using similar triangles. If he holds his 2-meter high staff perpendicular to the ground so that the end of the cliff's shadow coincides with the end of the staff's shadow, what is the height of the cliff?
(1) The staff's shadow is 5 meters long.
(2) The cliff's shadow is 10 times as long as the staff's shadow.
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03 Nov 2021, 06:41
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This question is a part of Are You Up For the Challenge: 700 Level Questions collection.
15 Dec 2021, 02:38
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Call the cliff's height = H
The staff's height = 2
the Shadow from the bottom far left of the figure up to the point of the staff's intersection ("cliff's shadow") = C
the shadow that the staff makes - from the point of the staff's intersection to the far bottom right-end of the figure = S
The smaller right triangle is Similar to the Entire Triangular Figure - All three Angles are Congruent (A-A-A Postulate)
accordingly, the corresponding lengths for each triangle that is across from the congruent angle will be proportional:
2 / S = H / (C + S)
what is H = ?
s1: gives us the staff's shadow as 5 meters long
S = 5
2/5 = H / (C + 5)
still not enough information to determine a unique value for H = height of cliff
NOT Sufficient
S2: the cliff's shadow is 10 times the length of the staff's shadow
C = 10(S) ---> plugging these values into the Given Proportion we set up:
2 / S = H / (C + S)
2 / S = H / (10S + S)
2 / S = H / 11S ---> the Variable S cancels in the DEN on both sides of the Proportion
2 / 1 = H / 11 --> H = 22 meters
-B- S2 Sufficient alone to get a unique value for the Height of the Cliff
The staff's height = 2
the Shadow from the bottom far left of the figure up to the point of the staff's intersection ("cliff's shadow") = C
the shadow that the staff makes - from the point of the staff's intersection to the far bottom right-end of the figure = S
The smaller right triangle is Similar to the Entire Triangular Figure - All three Angles are Congruent (A-A-A Postulate)
accordingly, the corresponding lengths for each triangle that is across from the congruent angle will be proportional:
2 / S = H / (C + S)
what is H = ?
s1: gives us the staff's shadow as 5 meters long
S = 5
2/5 = H / (C + 5)
still not enough information to determine a unique value for H = height of cliff
NOT Sufficient
S2: the cliff's shadow is 10 times the length of the staff's shadow
C = 10(S) ---> plugging these values into the Given Proportion we set up:
2 / S = H / (C + S)
2 / S = H / (10S + S)
2 / S = H / 11S ---> the Variable S cancels in the DEN on both sides of the Proportion
2 / 1 = H / 11 --> H = 22 meters
-B- S2 Sufficient alone to get a unique value for the Height of the Cliff
