the bases of a trapezoid are 22 and 12 respectively. The angles at the extremities of one base are 65^ and 40^ respectively. Find the two legs
Since the long base is 10 more than the short one, if you eliminate the interior rectangle of side 12, you have two triangles whose bases add up to 10.
You know two of the angles, so the 3rd angle is 75°. Now just use the law of sines to find the other two sides (the legs of the trapezoid):
10/sin75° = a/sin65° = b/sin40°
Let the trapezoid ABCD be of height h.
Given:
AB = 22
CD = 12
DAB = 65°
ABC = 40°
Construct a diagram and consider the trigonometry:
(1) AB = CD + h cot(DAB) + h cot(ABC)
(2) BC = h sec(ABC)
(3) DA = h sec(DAB)
Substitute the values into (1) to solve for h.
Substitute h into (2) and (3) to find the lengths of the other two sides.
To find the two legs of a trapezoid, we can use the properties of trapezoids and the given information.
First, let's label the given information:
- Base 1: 22 units
- Base 2: 12 units
- Angle at one end of Base 1: 65 degrees
- Angle at the other end of Base 1: 40 degrees
In a trapezoid, the opposite angles (angles at the extremities of the same base) are congruent. Therefore, the angles at the extremities of Base 2 will be 65 degrees and 40 degrees as well.
To find the two legs, we can use the trigonometric functions sine and cosine.
Step 1: Find the length of the shorter leg adjacent to the 40-degree angle.
- Let's label this leg as leg 1.
- We can use the sine function because we have the opposite (height) side and the hypotenuse (base) side.
- We can use the formula: sin(angle) = opposite/hypotenuse.
- Plugging in the values, we get: sin(40 degrees) = leg 1 / 12.
- Rearranging the equation, we find: leg 1 = sin(40 degrees) * 12.
Step 2: Find the length of the longer leg adjacent to the 65-degree angle.
- Let's label this leg as leg 2.
- We can use the cosine function because we have the adjacent side (leg 2) and the hypotenuse (base) side.
- We can use the formula: cos(angle) = adjacent/hypotenuse.
- Plugging in the values, we get: cos(65 degrees) = leg 2 / 22.
- Rearranging the equation, we find: leg 2 = cos(65 degrees) * 22.
Using a calculator, we can evaluate sin(40 degrees) and cos(65 degrees):
- sin(40 degrees) ≈ 0.6428.
- cos(65 degrees) ≈ 0.4226.
Now we can substitute these values into the equations to find the lengths of the legs:
- leg 1 = 0.6428 * 12 ≈ 7.714 units.
- leg 2 = 0.4226 * 22 ≈ 9.298 units.
Therefore, the length of the shorter leg (leg 1) is approximately 7.714 units, and the length of the longer leg (leg 2) is approximately 9.298 units.