AB ∥ DC, DE ⊥ AB , AD = BC, m∠ADC = 134° AD = 40, DC = 32, ABCD is a trapezoid. Find: Area of ABCD
To find the area of the trapezoid ABCD, we can divide it into two triangles and a rectangle.
Step 1: Find the height of the trapezoid.
Since DE is perpendicular to AB, it is also perpendicular to DC. Therefore, triangle ADE is a right triangle. We can use the Pythagorean theorem to find the height DE.
Using the Pythagorean theorem:
DE^2 + AE^2 = AD^2
DE^2 + (BC + AE)^2 = AD^2
DE^2 + (32 + AE)^2 = 40^2
We know that AD = BC = 40, and DC = 32.
Substituting these values:
DE^2 + (32 + AE)^2 = 40^2
DE^2 + (32 + AE)^2 = 1600
Step 2: Solve for the height DE.
Since AB is parallel to DC, triangle ACD and triangle ABE are similar triangles. So, we can set up a proportion to solve for AE.
DE/32 = AE/BC
DE/32 = AE/40
DE = (32/40) * AE
DE = (4/5) * AE
Substituting this value in the previous equation:
((4/5) * AE)^2 + (32 + AE)^2 = 1600
(16/25) * AE^2 + (32 + AE)^2 = 1600
Step 3: Solve for AE.
Solving the equation by simplifying and rearranging terms:
(16/25) * AE^2 + (AE + 32)^2 = 1600
16AE^2/25 + AE^2 + 64 + 64AE + AE^2 = 1600
16AE^2 + 25AE^2 + 25(64) + 25(64)AE + 25AE^2 = 1600 * 25
16AE^2 + 25AE^2 + 1600 + 1600AE + 25AE^2 = 40000
66AE^2 + 1600AE + 3200 = 40000
66AE^2 + 1600AE - 36800 = 0
This is a quadratic equation in terms of AE. We can solve it using factoring, completing the square, or the quadratic formula.
After solving for AE, we can substitute this value back into the equation DE = (4/5) * AE to find the value of DE.
Step 4: Calculate the area of the trapezoid.
The area of the trapezoid ABCD is given by the formula:
Area = (1/2)(sum of the bases)(height)
Area = (1/2)(AB + DC)(DE)
Substitute the values of AB, DC, and DE that you have found, and calculate the area.
To find the area of trapezoid ABCD, we can use the formula:
Area = (b1 + b2) * h / 2
where b1 and b2 are the lengths of the parallel bases, and h is the height.
In this case, AB is parallel to CD, so b1 = AD and b2 = BC. We are given that AD = BC, so we can substitute AD for b1 and BC for b2.
Now, let's find the height h of the trapezoid using triangle ADC:
We are given that DE is perpendicular to AB. This means that DE is the height of triangle ADC.
To find DE, we can use the Pythagorean theorem on triangle ADE:
AD^2 = AE^2 + DE^2
Substituting the given values, we have:
40^2 = AE^2 + DE^2
Now, we need to find AE. Since AB is parallel to CD, we can use the alternate interior angles theorem to find that angle ADE is congruent to angle ABC. Thus, triangle ABC is similar to triangle ADC.
Because the ratios of corresponding lengths in similar triangles are equal, we have:
AE / AD = AB / DC
Solving for AE, we get:
AE = (AB / DC) * AD
Substituting the given values, we have:
AE = (AB / 32) * 40
Since we know that AB is parallel to CD, we can also use the alternate interior angles theorem to find that angle ABC is congruent to angle ADC. This means that triangle ABC is isosceles, so AB = BC.
Now, we can substitute the values into the formula for the area of the trapezoid:
Area = (b1 + b2) * h / 2
= (AD + BC) * DE / 2
= (40 + 40) * DE / 2
= 80 * DE / 2
= 40 * DE
So, to find the area of trapezoid ABCD, we need to find DE and multiply it by 40.
m∠DAE = 180-134 = 46
sin 46 = DE/40
DE =40 sin 46
area = 32 * 40 sin 46