You have a triangle that measures 180 degrees. AB is 4 units. the measure of BC is 5 units. The measure of CD is 12 units. What is the measure of DE and how did you get that answer?
Your question is not clear to me.
The statement "You have a triangle that measures 180 degrees. " makes no sense, every triangle has an angle sum of 180°.
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I need help with definding classify
180
To find the measure of DE, we need to determine the length of AC first.
Given that the sum of angles in a triangle is always 180 degrees, we know that the sum of angles ABC and BCD must equal 180 degrees.
To find angle BCD, we can use the Law of Cosines, which states that in a triangle:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c represents the side opposite angle C, and a and b represent the other two sides.
In this case, angle BCD is angle C, side a is 5 units, side b is 12 units, and side c is DE (which we want to find).
Using the Law of Cosines, we can calculate the measure of angle C:
12^2 = 5^2 + DE^2 - 2 * 5 * DE * cos(C)
144 = 25 + DE^2 - 10DE * cos(C)
Rearranging the equation, we get:
DE^2 - 10DE * cos(C) = 144 - 25
DE^2 - 10DE * cos(C) = 119
Now, let's use the given information. We know that side AB measures 4 units, which means side AC is 4 + 5 = 9 units (since BC is 5 units).
Using the Law of Cosines again, we can find cos(C) as follows:
9^2 = 4^2 + 5^2 - 2 * 4 * 5 * cos(C)
81 = 16 + 25 - 40 * cos(C)
Rearranging the equation:
40 * cos(C) = 41
cos(C) = 41/40
Now we can substitute cos(C) into the equation for DE:
DE^2 - 10DE * (41/40) = 119
Multiplying each term by 40 to simplify:
40 * DE^2 - 410 * DE = 4760
Rearranging the equation:
40 * DE^2 - 410 * DE - 4760 = 0
Now, this equation is a quadratic equation in terms of DE. We can solve this equation using the quadratic formula:
DE = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 40, b = -410, and c = -4760. Plugging in these values, we get:
DE = (-(-410) ± √((-410)^2 - 4 * 40 * (-4760))) / (2 * 40)
DE = (410 ± √(168,100 - (-60,640))) / 80
Simplifying further:
DE = (410 ± √(228,740)) / 80
DE = (410 ± 478.22) / 80
Therefore, the two possible values for DE are:
DE = (410 + 478.22) / 80 ≈ 11.73
DE = (410 - 478.22) / 80 ≈ -0.98
Since a length cannot be negative, the measure of DE is approximately 11.73 units.