In the given figure, the length of BC is 6.66 cm and the length of DC is 5.33 cm
Trigonometry
From the question, we are to determine the length of BC and DC
Considering the diagram,
First, we will determine the measure of angle A
Using SOH CAH TOA
We can write that
[tex]sin A = \frac{4}{5}[/tex]
sin A = 0.8
∴ A = sin⁻¹(0.8)
A = 53.13°
Now, we will determine the measure of angle C
∠A + ∠ABC + ∠C = 180° (Sum of angles in a triangle)
53.13° + 90° + ∠C = 180°
∠C = 180° - 90° - 53.13°
∠C = 36.87°
Also,
Using SOH CAH TOA
[tex]tan C = \frac{BD}{DC}[/tex]
[tex]tan36.87 ^\circ=\frac{4}{DC}[/tex]
[tex]0.7500 = \frac{4}{DC}[/tex]
[tex]DC = \frac{4}{0.75}[/tex]
DC = 5.33 cm
By the Pythagorean theorem,
/BC/² = /BD/² + /DC/²
/BC/² = 4² + 5.33²
/BC/² = 16 + 28.4089
/BC/² = 44.4089
/BC/ = √44.4089
/BC/ = 6.66 cm
Hence, the length of BC is 6.66 cm and the length of DC is 5.33 cm
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