In triangle ABC angle BAC=90, angle ABC: angle ACB=1:2 and AC = 4cm. Calculate the length of BC.

reevelingw97

reevelingw97

Answered question

2022-11-03

Hypotenuse and angle ratio relationship
In triangle B A C = 90, A B C: A C B = 1 : 2 and A C = 4 c m. Calculate the length of BC. I am interested in the solution using Pythagoras theorem.

Answer & Explanation

gortepap6yb

gortepap6yb

Beginner2022-11-04Added 19 answers

Step 1
Well if B A C = 90 and A B C : A C B = 1 : 2, this means A B C = 60 and A C B = 30 .
Step 2
Now A C = 4 and B C = A B 2 + A C 2 . But A B = B C cos ( 60 ).
anraszbx

anraszbx

Beginner2022-11-05Added 2 answers

Step 1
B A C = 90 o , A B C : A C B = 1 : 2 , A C = 4
Draw a line through A such that B A D = 60 o . Let it intersect BC at D. Now ADB is an equilateral triangle. So:
A D = D B = B A , A D B = D B A = B A D = 60 o

As A D C = 180 0 A D B = 120 o
In A C D
D A C = 180 o A C D C D A = 180 o 30 o 120 o = 30 o
So A C D = C A D, so sides A D = C D.
Now let B A = x, so A B = B D = D A = x A D = C D = x
Step 2
Apply pythagoreas theorem in A B C:
A B 2 + A C 2 = B C 2
So: x 2 + A C 2 = ( 2 x ) 2 = 4 x 2 A C 2 = 3 x 2 A C = 3 x
As: A C = 3 x = 4 x = 4 / 3
Hence B C = 2 x = 8 / 3

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