A semicircle has AB as the endpoints of its diameter, and AB=80. Points C and D are on the circumference of the semicircle such that AD=BC=40. What is the length of DC?
Since the radius is equal to the given chords, the central angle is 60 degrees.
(An inscribed hexagon has sides equal to the radius)
So, DC is also 40
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To find the length of DC, we need to understand the properties of a semicircle and apply the relevant formulas.
In a semicircle, the diameter is the longest chord and passes through the center of the circle. The radius is the distance from the center of the semicircle to any point on its circumference.
In this problem, we are given that AB is the diameter of the semicircle and its length is 80. Therefore, the radius is half the length of the diameter, which is 80/2 = 40.
Since AD = BC = 40, we can conclude that AC and BD are also radii of the semicircle. Hence, AC and BD both have a length of 40 units.
Now, let's consider triangle DCA.
We know that AD = AC = 40, and since line DC is a chord of the semicircle, it is not a radius. Therefore, DC is not equal to 40.
To find the length of DC, we can use the Pythagorean theorem applied to triangle DCA. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In triangle DCA, DC is the hypotenuse, and DA and AC are the other two sides. So, we can use the Pythagorean theorem:
DC^2 = DA^2 + AC^2
Since DA and AC are both equal to 40, we can substitute these values into the equation:
DC^2 = 40^2 + 40^2
Simplifying further:
DC^2 = 1600 + 1600
DC^2 = 3200
To find the length of DC, we take the square root of both sides of the equation:
DC = √3200
Using a calculator, we find that the square root of 3200 is approximately 56.57 (rounded to two decimal places).
Therefore, the length of DC is approximately 56.57 units.