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 = 5, angleA = 40°
See 4-13-11, 12:32am post for solution.
c squared= a squared + b squared
c squared= 5 squared + 6 squared
c squared= 25 + 36
c = square root of 61
c = 7.81
To find the length of the diagonals AC and BD, we can use the law of cosines.
Step 1: Determine the lengths of AB, BC, CD, and AD. We are given AB = DC = 6 and AD = BC = 5.
Step 2: Determine the measure of angle B. Since a parallelogram has opposite angles equal, we have angle B = angle D = 180 - angle A = 180 - 40 = 140°.
Step 3: Apply the law of cosines to find the length of diagonal AC.
In triangle ABC, we have:
AC^2 = AB^2 + BC^2 - 2*AB*BC*cos(angle B)
AC^2 = 6^2 + 5^2 - 2*6*5*cos(140°)
Step 4: Calculate the value of AC^2.
AC^2 = 36 + 25 - 60*cos(140°)
Step 5: Evaluate the cosine of 140°.
cos(140°) ≈ -0.766
Step 6: Substitute the value of cos(140°) and calculate AC^2.
AC^2 = 36 + 25 - 60*(-0.766)
AC^2 = 61 + 45.96 ≈ 106.96
Step 7: Take the square root of AC^2 to find the length of AC.
AC ≈ √106.96 ≈ 10.34
Therefore, the length of diagonal AC is approximately 10.34.
Step 8: Repeat steps 3-7 to find the length of diagonal BD.
In triangle ABD, we have:
BD^2 = AD^2 + AB^2 - 2*AD*AB*cos(angle A)
BD^2 = 5^2 + 6^2 - 2*5*6*cos(40°)
Calculate BD^2 using the same procedure as before, and then take the square root to find the length of BD.
BD ≈ √(value of BD^2)
Round the answer to two decimal places.