solve the triangle
triangle BCD <D=90 degrees <B=21 degrees b=4
show all work and steps!
To solve the triangle BCD, we are given that angle D is 90 degrees, angle B is 21 degrees, and side b is 4. We need to find the other angles and sides of the triangle.
First, let's label the triangle with the given information:
C
/|
/ |
b / |a
/ |
/ |
D-----B
We know that angle D is 90 degrees, so we have:
C
/|
/ |
b / |
/ |
/ |
|-----B--D
Next, let's use the Angle Sum Property of a triangle to find angle C:
Angle C = 180 degrees - (Angle D + Angle B)
Angle C = 180 degrees - (90 degrees + 21 degrees)
Angle C = 180 degrees - 111 degrees
Angle C = 69 degrees
Now, we have:
C
/|
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b / | a
/ |
/ |
|----B--D
Next, we will find side a using the Sine Law:
sin(B) / b = sin(C) / a
Substitute the given values:
sin(21) / 4 = sin(69) / a
Cross-multiply:
sin(21) * a = 4 * sin(69)
Divide both sides by sin(21):
a = (4 * sin(69)) / sin(21)
a ≈ 7.14
Now, we have:
C
/|
/ |
7.14/ | 4
/ |
/ |
|----B--D
To summarize, the solution to the triangle BCD is:
Angle B = 21 degrees
Angle C = 69 degrees
Angle D = 90 degrees
Side b = 4
Side a ≈ 7.14