A parallelogram ABCD has lengths of sides and angles given below. Find the length of the diagonals AC and BD. (Round your answers to two decimal places.)
AB = DC = 6 and AD = BC = 3, A = 40°
To find the length of the diagonals AC and BD, we can use the law of cosines.
For diagonal AC:
Step 1: Find angle CAD:
Since A = 40°, angle CAD = 180° - A = 180° - 40° = 140°.
Step 2: Apply the law of cosines:
AC² = AD² + DC² - 2 x AD x DC x cos CAD.
Substituting the values:
AC² = 3² + 6² - 2 x 3 x 6 x cos 140°.
Step 3: Calculate AC:
AC = √(AC²).
Similarly, we can find the length of diagonal BD.
For diagonal BD:
Step 1: Find angle BDA:
Since angle BDA = 180° - angle CAD = 180° - 140° = 40°.
Step 2: Apply the law of cosines:
BD² = BA² + AD² - 2 x BA x AD x cos BDA.
Substituting the values:
BD² = 6² + 3² - 2 x 6 x 3 x cos 40°.
Step 3: Calculate BD:
BD = √(BD²).
Now, let's calculate AC and BD:
To find the length of the diagonals AC and BD in the given parallelogram, we can use the properties of a parallelogram.
1. First, let's draw the parallelogram and label the given information:
A_________B
/ /
/ /
/ /
/ /
D_________C
Given information:
AB = DC = 6
AD = BC = 3
A = 40°
2. Since opposite sides of a parallelogram are equal in length, we can conclude that AC = BD.
3. To find the length of AC, we can use the law of cosines in triangle ABC:
AC² = AB² + BC² - 2 * AB * BC * cos(A)
AC² = 6² + 3² - 2 * 6 * 3 * cos(40°)
AC² = 36 + 9 - 36 * cos(40°)
4. Calculate the value of AC²:
AC² = 36 + 9 - 36 * cos(40°)
AC² = 36 + 9 - 36 * 0.766
AC² ≈ 36 + 9 - 27.576
AC² ≈ 17.424
5. Finally, find the length of AC:
AC ≈ √(17.424) ≈ 4.17 (rounded to two decimal places)
6. Since AC = BD, the length of BD is also approximately 4.17.
Therefore, the length of the diagonals AC and BD in the parallelogram ABCD is approximately 4.17 units.
A = C = 40deg.
B = D = (360-80) / 2 = 140 deg.
Using the Law of signs,
BD / sinA = AB / sin(D/2),
BD / sin40 = 6 / sin70,
Multiply both sides by sin40:
BD = 6sin40 / sin70 = 4.1.
AC / sinD = AD / sin(C/2),
AC / sin140 = 3 / sin20,
AC = 3sin140 / sin20 = 5.64.