ad=10 units;ab=8 units;ac=12 units;ed=4.5 units.if <bad=50,what is the measure of<adc?
To determine the measure of ∠ADC, we can use the Law of Cosines. The formula is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we have:
c = CD = ed = 4.5 units
a = AD = 10 units
b = AC = 12 units
∠C = ∠BAD = 50°
Now, let's substitute the values into the formula:
CD^2 = AD^2 + AC^2 - 2 * AD * AC * cos(∠C)
4.5^2 = 10^2 + 12^2 - 2 * 10 * 12 * cos(50°)
20.25 = 100 + 144 - 240 * cos(50°)
20.25 = 244 - 240 * cos(50°)
240 * cos(50°) = 244 - 20.25
240 * cos(50°) = 223.75
cos(50°) = 223.75 / 240
Now, calculate the inverse cosine to find the angle:
∠ADC = 1 / cos^-1(223.75 / 240)
∠ADC ≈ 26.17°
Therefore, the measure of ∠ADC is approximately 26.17 degrees.
To find the measure of ∠ADC, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, c and opposite angles A, B, C, the following equation holds true:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, we want to find the measure of angle ∠ADC, which is opposite side ac, with lengths ab = 8 units and ed = 4.5 units. We are given that angle ∠BAD = 50°.
First, let's calculate the length of side bd using the Law of Cosines:
bd^2 = ab^2 + ed^2 - 2ab*ed*cos(BAD)
bd^2 = 8^2 + 4.5^2 - 2*8*4.5*cos(50°)
bd^2 = 64 + 20.25 - 2*8*4.5*cos(50°)
bd^2 = 144.25 - 72*cos(50°)
Now, we can use the Law of Cosines again to find the measure of angle ∠ADC:
ac^2 = ad^2 + cd^2 - 2ad*cd*cos(ADC)
We know that ac = 12 units, ad = 10 units, and we want to find cd.
cd^2 = ac^2 + ad^2 - 2ac*ad*cos(ADC)
cd^2 = 12^2 + 10^2 - 2*12*10*cos(ADC)
cd^2 = 144 + 100 - 240*cos(ADC)
cd^2 = 244 - 240*cos(ADC)
Now we have two equations:
bd^2 = 144.25 - 72*cos(50°)
cd^2 = 244 - 240*cos(ADC)
To find the measure of angle ∠ADC, we need to solve for cd by substituting the value of bd in the second equation:
cd^2 = 244 - 240*cos(ADC)
cd^2 = 244 - 240*cos(50°)
Now, we can solve for cd by taking the square root of both sides:
cd = √(244 - 240*cos(50°))
Calculating this value will give you the length of cd.