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Triangle Distances Puzzle – Thoughts Your Selections

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 typical, watch the video for an answer.

Triangle Distances Puzzle

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“All will probably be properly should you use your thoughts to your selections, and thoughts solely your selections.” Since 2007, I’ve devoted my life to sharing the enjoyment of sport idea and arithmetic. MindYourDecisions now has over 1,000 free articles with no adverts because of group assist! Assist out and get early entry to posts with a pledge on Patreon.


Reply To Triangle Distances Puzzle

(Just about all posts are transcribed rapidly after I make the movies for them–please let me know if there are any typos/errors and I’ll appropriate them, thanks).

One technique 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 are able to then remedy ∠CPP’ = ∠CP’P = 60°, so triangle PCP’ is equilateral and PP’ = 6.

Let ∠BPP’ = θ In triangle BPP’ we’ll use Al-Kashi’s regulation 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’ll use we’ll use Al-Kashi’s regulation of cosines to get:

s2 = 52 + 62 – 2(5)(6) cos (θ + 60°)

We will use the cosine sum system 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 are able to calculate the realm of an equilateral triangle from its system:

Space ABC = (s2√3)/4
Space ABC = (55/4)√3 + 9√6 ≈ 45.861

Particular thanks this month to:

Mike Robertson
Michael Anvari

Because of all supporters on Patreon.


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