if segment AD is 10 units; segment AB is 8 units; segment AC is 12 units; segment ED is 4.5 units. If angle BAD is 50 degrees what is the measure of angle ADC
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To find the measure of angle ADC, we can use the Law of Cosines. The Law of Cosines states that in any triangle ABC, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those two sides and the cosine of the included angle.
In this case, we have triangle ADC with side lengths AD = 10 units, AC = 12 units, and angle ADC as the included angle. We are given the length of segment ED, which is 4.5 units, which can be considered as the third side of triangle AED.
Using the Law of Cosines, we can calculate the length of segment AE using the formula:
AE^2 = AD^2 + ED^2 - 2 * AD * ED * cos(BAD)
Plugging in the values:
AE^2 = 10^2 + 4.5^2 - 2 * 10 * 4.5 * cos(50°)
Now, we can solve for AE:
AE^2 = 100 + 20.25 - 90 * cos(50°)
AE^2 = 120.25 - 90 * cos(50°)
AE ≈ √(120.25 - 90 * cos(50°))
Once we have AE, we can apply the Law of Cosines again to find the angle ADC. The equation is:
cos(ADC) = (AC^2 + AD^2 - CD^2) / (2 * AC * AD)
Plugging in the given values:
cos(ADC) = (12^2 + 10^2 - AE^2) / (2 * 12 * 10)
Now, we can solve for cos(ADC):
cos(ADC) = (144 + 100 - AE^2) / 240
cos(ADC) = (244 - AE^2) / 240
Finally, we can find the angle ADC using the inverse cosine function (cos^-1) or arc cosine function:
ADC ≈ cos^-1((244 - AE^2) / 240)
Note: To get the final numerical value of angle ADC, you need to substitute the value of AE^2 that we calculated earlier.