riangular flower garden ABC in which AB = 4m, BC = 5and ∠BCA =30. Point D lies on AC such that BD = 4 m and ∠BDC is obtuse.
Find:
(a) ∠BDC
(b) the length of AD
(c) the length of DC
(d) the area of the flower garden ABC
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asked by King
Jan 10, 2020
If you made a reasonable sketch you will see that ∆ABD is isosceles and you are given that
∠BDC is obtuse.
Look at ∆BCD, you can use the sine law to find ∠BDC
sin D/5 = sin 30°/4
sinD = 5(1/2) / 4 = 5/8
D = 38.68°
So now you can find ∠BDA and ∠BAD and from there all remaining angles.
b) for AD, use the sine law
sinAD/4 = 4/sinD
c) use the sine law
5/sin30 =4/DC
d) area = (1/2)(5)(4)sin
I see one error in my previous post to this, and it needs one clarification
change the following
D = 38.68°
So now you can find ∠BDA and ∠BAD and from there all remaining angles.
to
D = 38.68° or 141.32° because the sine is positive in I or II.
But you said ∠BDC is obtuse , so make sure your diagram shows D as the obtuse angle
So now you can find ∠BAD and from there all remaining angles.
The error is here: it should be
b)for AD, use the sine law
AD/sin DBA = 4/sin38.68°
What about length of DC
(a) To find angle ∠BDC, we can use the sine law. In triangle BCD, we have the values BC = 5m and ∠BCA = 30°. From ∆BCD, we can set up the following equation using the sine law:
sin ∠BDC / 5 = sin 30° / 4
Rearranging the equation, we have:
sin ∠BDC = (5/2) * (1/4) = 5/8
Using inverse sine function, we can find ∠BDC:
∠BDC = sin^(-1)(5/8) ≈ 38.68°
Therefore, angle ∠BDC is approximately 38.68°.
(b) To find the length of AD, we can use the sine law again. In triangle ABD, we have the values AD (which we are trying to find) and BD = 4m. Using the angle ∠BDC we found in part (a), we can set up the following equation using the sine law:
sin ∠BAD / AD = sin ∠BDC / BD
Rearranging the equation, we have:
AD = BD * sin ∠BAD / sin ∠BDC
Substituting the known values, we get:
AD = 4 * sin ∠BAD / sin ∠BDC
(c) To find the length of DC, we can again use the sine law. In triangle BCD, we have the values BC = 5m and ∠BCA = 30°. We want to find DC. Using the sine law, we can set up the following equation:
5 / sin 30° = DC / sin ∠BDC
Rearranging the equation, we have:
DC = (5 * sin ∠BDC) / sin 30°
Substituting the known values, we get:
DC = (5 * sin ∠BDC) / (1/2)
(d) To find the area of the flower garden ABC, we can use the formula for the area of a triangle. Given the lengths of two sides and the included angle, we can calculate the area using the following formula:
Area = (1/2) * BC * AD * sin ∠BCA
Substituting the known values, we get:
Area = (1/2) * 5 * AD * sin 30°
Therefore, the area of the flower garden ABC is (1/2) * 5 * AD * sin 30°.