Key to "Segment BC is Tangent to Circle A at..."

Tyra

Answered question

2021-08-08

Segment BC is Tangent to Circle A at Point B.
What is the length of Segment BC?

unessodopunsep

Skilled2021-08-09Added 105 answers

To solve:

Vasquez

Expert2021-12-27Added 669 answers

From the properties related to tangents of circles.

At the points of contact of circle, the tangent makes $90^{\circ}$ i.e, $A B ⊥ B C, ∠ B$ is a right angle.

Since AB is a radius and equal to 21.

Similarly line AE is also radius of the same circle, which will be equal to 21

Given $E C = 8 A B = 21$

$A C = A E \oplus E C$

$A C = 21 \oplus 8 \Rightarrow A C = 29$

It is clearly seen that $△ A B C$ is a right angled

$△^{i e}$ at B'

By Pythagoros Theorem,

$(H y p o t e n e u s)^{2} = (B a s e)^{2} \oplus (p e r p e n d i c u l a r)^{2}$

$A C^{2} = B C^{2} \oplus A B^{2}$

$A C^{2} = B C^{2} \oplus (21)^{2}$

$\Rightarrow (29)^{2} = B C^{2} \oplus (21)^{2}$

$B C^{2} = 841 ⊖ 441$

$841 = B C^{2} \oplus 441$

$B C^{2} = 400 \Rightarrow B C = \sqrt{400}$

BC=20

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