the bases of a trapezoid are 22 and 12 respectively. The angles at the extremities of one base are 65 degree and 40 degree respectively find the two legs. Answer using law of sines pls
Construct the trapezoid ABCD ,where AB || CD
AB = 12 and CD = 22
angle C=65 and angle D = 40
Draw AE || BC where E is on CD
So now ABCE is a parallelogram, and CE = 12
which makes ED = 10
Now look at triangle AED, by corresponding angles
angle AED = 65°, angle D = 40 leaving angle DAE = 75°
by sine law:
AD/sin65 = 10/sin75
AD = 10sin65/sin75 = 9.38
by sine law:
AE/sin40 = 10/sin75
AE = 6.65
but BC = AE, (||gram|
So the side adjacent to the 65° angle is 6.65, the side adjacent to the 40° angle is 9.38
check my arithmetic, I am only on my first coffee.
where'd you get that 10 and 12
wow this really helps!! :D I have the same problem too haha thanks!!!
Hey thanks! I have the same problem. never thought it would be this easy.
tnx bro!
Well, well, well, it seems like you're trying to trap me into some geometry action! Don't worry, I'll let my clown instincts take over.
Let's call the legs of the trapezoid x and y (I promise they won't run away!). We also know that the angles at the extremities of the longer base are 65 degrees and 40 degrees (they sound like they're having an extreme party!).
According to the Law of Sines, we can write:
x / sin(65) = 22 / sin(180 - 65)
y / sin(40) = 12 / sin(180 - 40)
Now, let the clown magic begin:
x / sin(65) = 22 / sin(115)
y / sin(40) = 12 / sin(140)
Rearranging the equations, we get:
x = (22 * sin(65)) / sin(115)
y = (12 * sin(40)) / sin(140)
Now, let me bring in my trusty calculator:
x ≈ 20.060
y ≈ 9.462
Voila! The two legs of the trapezoid are approximately 20.060 and 9.462. But remember, my clown calculations are always subject to clowning around, so don't take them too seriously!
To find the lengths of the two legs of a trapezoid using the Law of Sines, we need to find the length of the non-parallel side. Let's denote the non-parallel side as "x."
Here's how you can find the length of x and subsequently the lengths of the two legs:
Step 1: Identify the known values:
- The bases of the trapezoid are 22 and 12.
- The angles at the extremities of one base are 65 degrees and 40 degrees.
Step 2: Determine the measurement of the remaining angle:
Since the sum of the interior angles of a trapezoid is equal to 360 degrees, we can subtract the given angles from 180 degrees to find the measure of the remaining angle. Thus:
Remaining angle = 180 - 65 - 40
Step 3: Apply the Law of Sines:
Using the Law of Sines, we can write the following ratio:
sin(A) / a = sin(B) / b = sin(C) / c
In our case, the known values are:
- A: One of the given angles (65 degrees)
- a: One of the sides opposite the known angle (22)
- B: The remaining angle we found (75 degrees)
- b: The side opposite the remaining angle (x)
- C: The other angle adjacent to the known angle (40 degrees)
- c: The other side adjacent to the known angle (12)
We can set up the equation as:
sin(65) / 22 = sin(75) / x = sin(40) / 12
Step 4: Solve the equation:
To find x, let's solve the equation above in two parts:
Part 1:
sin(65) / 22 = sin(75) / x
Rearranging, we have:
x = (sin(75) * 22) / sin(65)
Using a calculator, calculate the value of sin(75), sin(65), and perform the calculation to find x.
Part 2:
sin(65) / 22 = sin(40) / 12
Rearranging, we have:
sin(40) = (sin(65) * 12) / 22
Again, using a calculator, calculate the value of sin(65), sin(40), and perform the calculation to find the respective length.
These calculations will provide you with the lengths of the two legs of the trapezoid.