triangular 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
calling tbhe 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
To find the values requested, we can utilize the given information and apply relevant geometric principles. Let's solve each part step-by-step:
(a) To find ∠BDC, we'll first need to calculate ∠ABC using the Law of Cosines for triangle ABC:
cos(∠ABC) = (AB^2 + BC^2 - AC^2) / (2 * AB * BC)
cos(∠ABC) = (4^2 + 5^2 - AC^2) / (2 * 4 * 5)
cos(∠ABC) = (16 + 25 - AC^2) / 40
cos(∠ABC) = (41 - AC^2) / 40
Since ∠BCA = 30 degrees, we know that ∠ABC = 180 - ∠BCA = 180 - 30 = 150 degrees. Using this information, we can calculate cos(∠ABC):
cos(∠ABC) = cos(150)
cos(∠ABC) = -√3/2
Now, let's substitute this value back into the previous equation:
-√3/2 = (41 - AC^2) / 40
To solve for AC^2, we can cross-multiply and simplify:
-√3 * 40 = 41 - AC^2
-√3 * 40 - 41 = -AC^2
-√3 * 40 - 41 = AC^2
AC^2 = (√3 * 40) + 41
AC^2 = 68.881
Taking the positive square root of both sides, we find:
AC = √68.881
AC ≈ 8.3m
Now, we have three sides of triangle BDC: BD = 4m, DC = AC - AD, and BC = 5m. We want to find ∠BDC.
Using the Law of Cosines for triangle BDC:
cos(∠BDC) = (BD^2 + DC^2 - BC^2) / (2 * BD * DC)
cos(∠BDC) = (4^2 + (AC - AD)^2 - 5^2) / (2 * 4 * (AC - AD))
Since we obtained the value of AC earlier, we can substitute it into the equation and solve for ∠BDC using the given information.
(b) To find the length of AD, we can set up a right triangle ABD with BD = 4m and ∠BDA = 90 degrees. Using the Pythagorean theorem:
AD^2 = AB^2 - BD^2
AD^2 = 4^2 - 4^2
AD^2 = 16 - 16
AD^2 = 0
AD = √0
AD = 0m
(c) To find the length of DC, we can subtract the length of AD from AC:
DC = AC - AD
DC = 8.3m - 0m
DC = 8.3m
(d) To find the area of triangle ABC, we can use the formula for the area of a triangle given two sides and the included angle:
Area = (1/2) * AB * BC * sin(∠ABC)
Plugging in the values we know:
Area = (1/2) * 4m * 5m * sin(150)
Area = (1/2) * 20m^2 * (√3/2)
Area = 10m^2 * (√3/2)
Area = 5√3 m^2
Thus, the area of the flower garden ABC is approximately 5√3 square meters.
To solve this problem, we can use the properties of triangles and trigonometry. Let's break the problem down step-by-step:
(a) To find ∠BDC, we need to consider the triangle BDC. We know that BD = 4m, and we need to find ∠BDC. Since we are given that ∠BCA = 30° and ∠BDC is obtuse, we can conclude that ∠BDC = 180° - ∠BCA. Substituting the given angle value, we have:
∠BDC = 180° - 30°
∠BDC = 150°
(b) To find the length of AD, we can consider right triangle ABD. We know that AB = 4m and BD = 4m. Applying the Pythagorean theorem, we can find the length of AD:
AD² = AB² - BD²
AD² = (4m)² - (4m)²
AD² = 16m² - 16m²
AD² = 0
AD = 0m
Therefore, the length of AD is 0m.
(c) To find the length of DC, we can consider triangle BDC. We are given that BD = 4m, and we know that ∠BDC = 150°. We can use the law of cosines to find DC:
DC² = BD² + BC² - 2 * BD * BC * cos(∠BDC)
DC² = (4m)² + (5m)² - 2 * (4m) * (5m) * cos(150°)
DC² = 16m² + 25m² - 40m² * cos(150°)
DC² = 241m² + 40m² * cos(150°)
To calculate cos(150°), we can use the fact that cos(180° - x) = -cos(x). Thus, cos(150°) = -cos(30°). Since cos(30°) = √3/2, we have:
DC² = 241m² + 40m² * (-√3/2)
DC² = 241m² - 40m² * √3/2
DC² = 241m² - 20√3m²
Taking the square root on both sides, we get:
DC = √(241m² - 20√3m²)
Therefore, the length of DC is approximately √(241m² - 20√3m²)m.
(d) To find the area of the flower garden ABC, we need to calculate the area of triangle ABC. We can use the formula for the area of a triangle:
Area = 1/2 * base * height
Since triangle ABC is a right triangle with AC as the base and AB as the height, we have:
Area = 1/2 * AC * AB
Area = 1/2 * 5m * 4m
Area = 10m²
Therefore, the area of the flower garden ABC is 10m².