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Place coordinates:
𝐵 ( 0 , 0 ) , 𝐴 ( 𝑎 , 0 ) , 𝐶 ( 0 , 𝑏 ) B(0,0),A(a,0),C(0,b).
Midpoint
( 𝑎 / 2 , 𝑏 / 2 ) M=(a/2,b/2).
Form vectors:
𝐵 𝑀
( 𝑎 / 2 , 𝑏 / 2 ) BM =(a/2,b/2)
𝐵 𝐶
( 0 , 𝑏 ) BC =(0,b)
Dot product formula:
cos
𝐵 𝑀 ⃗ ⋅ 𝐵 𝐶 ⃗ ∣ 𝐵 𝑀 ⃗ ∣ ∣ 𝐵 𝐶 ⃗ ∣ cosθ= ∣ BM ∣∣ BC ∣ BM ⋅ BC
Simplify:
𝑏 𝑎 2 + 𝑏 2 cosθ= a 2 +b 2
b
Find angle:
cos − 1 ( 𝐵 𝐶 𝐴 𝐵 2 + 𝐵 𝐶 2 ) θ=cos −1 ( AB 2 +BC 2
BC
)
Round to nearest integer if needed.
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Find Angle MBC
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Place coordinates:
𝐵 ( 0 , 0 ) , 𝐴 ( 𝑎 , 0 ) , 𝐶 ( 0 , 𝑏 ) B(0,0),A(a,0),C(0,b).
Midpoint
𝑀
( 𝑎 / 2 , 𝑏 / 2 ) M=(a/2,b/2).
Form vectors:
𝐵 𝑀
⃗
( 𝑎 / 2 , 𝑏 / 2 ) BM =(a/2,b/2)
𝐵 𝐶
⃗
( 0 , 𝑏 ) BC =(0,b)
Dot product formula:
cos
𝜃
𝐵 𝑀 ⃗ ⋅ 𝐵 𝐶 ⃗ ∣ 𝐵 𝑀 ⃗ ∣ ∣ 𝐵 𝐶 ⃗ ∣ cosθ= ∣ BM ∣∣ BC ∣ BM ⋅ BC
Simplify:
cos
𝜃
𝑏 𝑎 2 + 𝑏 2 cosθ= a 2 +b 2
b
Find angle:
𝜃
cos − 1 ( 𝐵 𝐶 𝐴 𝐵 2 + 𝐵 𝐶 2 ) θ=cos −1 ( AB 2 +BC 2
BC
)
Round to nearest integer if needed.