The figure below represents a triangular flower garden ABC in which AB = 4m, BC = 5and ∠BCA =30 degrees . Point D lies on AC such that BD = 4 m and ∠BDC is obtuse.
Find, correct to 2 decimal places:
(a) the length of AD;
(b) the length of DC;
As I posted earlier,
calling the angle A,B,C as usual in a diagram, use the law of sines to find that
sin(∠BDC)/5 = sin(30)/4
so, sin ∠BDC = 5/8
But you want the obtuse angle.
Now you know that ∠BDA = 180 - ∠BDC
And, since triangle BDA is isosceles, ∠DAB = ∠BDA
Now you can find ∠B, and use the law of sines to find the other sides.
And recall that in any triangle ABC, the area = 1/2 ab sinC
So, where do you get stuck? Did you draw the triangle to get an idea what's going on?
Both. how to use law of sin
(a) The length of AD;
AD/4 =
(b) the length of DC
To find the length of AD and DC, we can use the Law of Cosines.
(a) To find the length of AD:
Let's label the angle at A as angle BAC.
Using the Law of Cosines:
AD^2 = AC^2 + CD^2 - 2(AC)(CD) * cos(angle BAC)
We know that AC = AB + BC = 4m + 5m = 9m
Since angle BAC is the supplement of angle BCA, which is 30 degrees, angle BAC = 180 degrees - 30 degrees = 150 degrees.
Substituting these values into the equation:
AD^2 = (9m)^2 + CD^2 - 2(9m)(CD) * cos(150 degrees)
(b) To find the length of DC:
Using the Law of Cosines again:
CD^2 = AD^2 + AC^2 - 2(AD)(AC) * cos(angle BAC)
Substituting the known values into the equation:
CD^2 = AD^2 + (9m)^2 - 2(AD)(9m) * cos(150 degrees)
Now we have two equations, one for AD^2 and one for CD^2. We can solve these equations simultaneously to find the values of AD and CD.
To find the lengths of AD and DC, we can use the law of cosines. The law of cosines states that for any triangle with sides a, b, and c and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
Let's apply the law of cosines to find the lengths of AD and DC:
(a) Length of AD:
In triangle ABC, we have AC = BC = 5m and ∠BCA = 30 degrees. We need to find the length of AD.
Applying the law of cosines to triangle ABC, we have:
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(∠BCA)
AC^2 = 4^2 + 5^2 - 2 * 4 * 5 * cos(30 degrees)
AC^2 = 16 + 25 - 40 * 0.866
AC^2 = 16 + 25 - 34.64
AC^2 = 6.36
Taking the square root of both sides, we get:
AC ≈ 2.52
Now, let's find the length of AD. Since BD = 4m, we can subtract BD from AC to get AD:
AD = AC - BD
AD = 2.52 - 4
AD ≈ -1.48
The length of AD is approximately -1.48m. However, since a negative length doesn't make sense in this context, it indicates that there may be an error or inconsistency in the given information or calculations. Please double-check the values and angles provided.
(b) Length of DC:
Since we were unable to find a valid length for AD in the given information, we won't be able to calculate the length of DC accurately.