To the nearest meter, find the length of side DE in triangle DEF, given angle F=54 degrees, side DF=70 m, and side EF=50 m.
Please show work!
cosine law:
DE^2 = 70^2 + 50^2 - 2(70)(50)cos54°
= ..
carry on
To find the length of side DE, we can use the Law of Cosines, which states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice their product multiplied by the cosine of the included angle.
Let's label side DE as x.
So, using the Law of Cosines, we have:
x^2 = 70^2 + 50^2 - 2 * 70 * 50 * cos(54)
To find the length of side DE, we need to solve for x. Let's calculate the right-hand side of the equation:
x^2 = 4900 + 2500 - 7000 * cos(54)
Using a calculator, we find that cos(54) is approximately 0.5878.
So, continuing the calculation, we have:
x^2 = 4900 + 2500 - 7000 * 0.5878
= 4900 + 2500 - 4106.6
= 3293.4
To find the value of x, we can take the square root of both sides:
x = √3293.4
Using a calculator, we find that √3293.4 is approximately 57.41.
Therefore, the length of side DE, to the nearest meter, is 57 meters.
To find the length of side DE, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we want to find the length of side DE, so we will use side DF as side a and side EF as side b. Let's substitute the given values into the equation:
DE^2 = 70^2 + 50^2 - 2 * 70 * 50 * cos(54)
Now we can calculate DE:
DE^2 = 4900 + 2500 - 7000 * cos(54)
Cos(54) ≈ 0.5878 (rounded to four decimal places)
DE^2 = 4900 + 2500 - 7000 * 0.5878
DE^2 = 4900 + 2500 - 4106.6
DE^2 ≈ 3293.4
To solve for DE, we need to take the square root of both sides:
DE ≈ √(3293.4)
DE ≈ 57.4 meters (rounded to the nearest meter)
Therefore, the length of side DE, to the nearest meter, is approximately 57 meters.