Question

ABC is a right angled triangle in which ∠ABC = 90° and ∠ACB = 60°. BC is produced to D such that ∠ADB = 45°. If CD = 30 cm, find the lengths of AB and BC. Please answer ASAP. First and best answer will be marked as BRAINLIEST.

Answer 1

Answer:

bc=41.09cm

ab=71.16788 cm

Answer 2

AB=$15(3+\sqrt 3)cm$

$BC=15(\sqrt 3+1)$cm

Step-by-step explanation:

Let AB=x and BC=y

Angle ACB=60 degrees

ADB=45 degrees

Angle ABC=90 degrees

In triangle ABD

$\frac{AB}{BD}=tan45$

By using formula $tan\theta=\frac{Perpendicular\;side}{base}$

$\frac{x}{y+30}=1$

$tan45=1$

$x=y+30$

In triangle ABC

$\frac{AB}{BC}=tan60$

$\frac{x}{y}=\sqrt 3$

$x=y\sqrt 3$

$y\sqrt 3=y+30$

$y\sqrt 3-y=30$

$y(\sqrt 3-1)=30$

$y=\frac{30}{\sqrt 3-1}\times \frac{\sqrt 3+1}{\sqrt 3+1}=\frac{30(\sqrt 3+1}{(\sqrt 3)^2-1)}=\frac{30(\sqrt 3+1)}{2}=15(\sqrt 3+1)$

By using formula:$(a+b)(a-b)=a^2-b^2$

$BC=15(\sqrt 3+1)$cm

Substitute the value

$x=15(\sqrt 3+1)\times \sqrt 3=15(3+\sqrt 3)cm$

AB=$15(3+\sqrt 3)cm$

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