Level P is within the inside of the equilateral triangle ABC. If AP = 7, BP = 5, and CP = 6, what’s the space of ABC?
As standard, watch the video for an answer.
Or preserve studying.
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“All might be properly if you happen to use your thoughts on your selections, and thoughts solely your selections.” Since 2007, I’ve devoted my life to sharing the enjoyment of recreation concept and arithmetic. MindYourDecisions now has over 1,000 free articles with no advertisements because of group help! Assist out and get early entry to posts with a pledge on Patreon.
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Reply To Triangle Distances Puzzle
(Just about all posts are transcribed shortly after I make the movies for them–please let me know if there are any typos/errors and I’ll appropriate them, thanks).
One option to remedy the issue is to “suppose outdoors the triangle.” Rotate APC 60 levels clockwise about C so it turns into BP’C.
We’ll first present ∠PCP’ = 60° as follows:
∠ACP = ∠BCP’ (by rotation)
∠PCP’ = ∠BCP’ + ∠PCB
∠PCP’ = ∠ACP + ∠PCB
∠PCP’ = ∠ACB
∠ACB = 60° (equilateral triangle)
∠PCP’ = 60°
Assemble PP’. This creates an isosceles triangle PCP’ with a vertex angle equal to 60°. As a result of PC = P’C = 6, we will then remedy ∠CPP’ = ∠CP’P = 60°, so triangle PCP’ is equilateral and PP’ = 6.
Let ∠BPP’ = θ In triangle BPP’ we are going to use Al-Kashi’s legislation of cosines to get:
72 = 52 + 62 – 2(5)(6) cos θ
cos θ = (52 + 62 – 72)/[2(5)(6)]
cos θ = 1/5
sin θ = √(1 – (cos θ)2)
sin θ = (2√6)/5
Suppose BC = s. Then in triangle BPC we are going to use we are going to use Al-Kashi’s legislation of cosines to get:
s2 = 52 + 62 – 2(5)(6) cos (θ + 60°)
We are able to use the cosine sum method to get:
cos (θ + 60°) = cos θ cos 60° – sin θ sin 60°
cos (θ + 60°) = (1/5)(1/2) – [(2√6)/5][(√3)/2)]
cos (θ + 60°) = (1 – 6√2)/10
Substituting to the equation for s2 we get:
s2 = 52 + 62 – 2(5)(6)(1 – 6√2)/10
s2 = 55 + 36√2
Then we will calculate the world of an equilateral triangle from its method:
Space ABC = (s2√3)/4
Space ABC = (55/4)√3 + 9√6 ≈ 45.861
Particular thanks this month to:
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Michael Anvari
Kyle
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Reference
Dialogue on Quora with many alternate options
https://www.quora.com/Inside-an-equilateral-triangle-there-is-a-point-which-is-5-6-and-7-units-away-from-the-3-vertices-respectively-What-is-this-triangles-area
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