The two non-parallel sides of an isosceles trapezoid are each 7 feet long. The longer of the two bases measures 22 feet long. The sum of the base angles is 140°.


a.
Find the length of the diagonal.
b.
Find the length of the shorter base.

Round your answers to the nearest hundredth.

Drop altitudes of length a from the ends of the shorter base. This gives you two right triangles at the ends, with altitude a and base x. Since the figure is isosceles, the two base angles are equal.

Now you have
Each base angle is 70°

a/7 = cos70°
x^2 + a^2 = 7^2

Now you know that if the shorter base is b,
2x+b = 22

the diagonal d is thus
d^2 = (x+b)^2 + a^2

To find the length of the diagonal and the shorter base of the isosceles trapezoid, we can use the properties of this type of trapezoid.

Given:
- The two non-parallel sides are each 7 feet long.
- The longer base measures 22 feet.
- The sum of the base angles is 140°.

a. Finding the length of the diagonal:
Step 1: Draw the isosceles trapezoid and label it.

B__________C
/ \
/ \
/ \
A/________________\D

AB = 7 ft
BC = 22 ft
CD = 22 ft
DA = 7 ft

∠BAD = ∠CDA = α
∠ABC = ∠DCB = β

Step 2: Notice that ∠ABC and ∠BCD are adjacent supplementary angles (they add up to 180°) since their sum is 140°.

∠ABC + ∠BCD = 180°
β + α = 180°

Step 3: We can write an equation using the fact that the opposite angles in a trapezoid are congruent (they have the same measure).

∠BAD = ∠CDA
α = ∠CDA

Step 4: Now we have two equations to work with:

α + β = 180° (equation 1)
α = ∠CDA (equation 2)

Step 5: Substitute equation 2 into equation 1 and solve for α.

α + β = 180°
∠CDA + β = 180°
α = 180° − β

Step 6: Substitute the value of α from equation 5 into equation 2 and solve for β.

∠CDA = 180° − β

∠CDA = α

So, 180° − β = α

Substituting,
180° − β = 180° − β

Since both sides are equal, we can conclude that β = β.

Step 7: Now we know that α = 180° − β and β = β. This means that α and β are congruent angles.

Step 8: From step 7, we can conclude that α = β.

So, α = β = x (let's assume both angles have the same measure x).

Step 9: Since the sum of the base angles (∠ABC + ∠DCB) is 140°, we can write:

2x = 140°
x = 70°

Step 10: Now, we know that ∠BAD = ∠CDA = α = β = x = 70°.

Step 11: Using the law of cosines, we can find the length of the diagonal DC.

Applying the law of cosines to triangle BCD, we have:

DC² = BC² + BD² - 2 * BC * BD * cos(∠CDB)

The length of BD is equal to the length of the shorter base, which we need to find.

Step 12: Now we can find the value of BD using the law of cosines.

BD² = BC² + CD² - 2 * BC * CD * cos(∠BCD)

BD² = 22² + 22² - 2 * 22 * 22 * cos(70°)
BD² = 484 + 484 - 2 * 22 * 22 * cos(70°)
BD² ≈ 369.237

Taking the square root of both sides gives us:

BD ≈ √369.237
BD ≈ 19.22 ft (rounded to the nearest hundredth)

Step 13: Now we can find the length of the diagonal DC using the law of cosines.

DC² = BC² + BD² - 2 * BC * BD * cos(∠CDB)

DC² = 22² + 19.22² - 2 * 22 * 19.22 * cos(70°)
DC² = 484 + 369.237 - 2 * 22 * 19.22 * cos(70°)
DC² ≈ 627.375

Taking the square root of both sides gives us:

DC ≈ √627.375
DC ≈ 25.05 ft (rounded to the nearest hundredth)

Therefore, the length of the diagonal is approximately 25.05 feet.

b. Finding the length of the shorter base:

Step 14: We know that the length of the shorter base BD is approximately 19.22 feet (as calculated in step 12).

Therefore, the length of the shorter base is approximately 19.22 feet.

To find the length of the diagonal and the shorter base of the isosceles trapezoid, we can use the properties and formulas of a trapezoid.

a. Find the length of the diagonal:

In an isosceles trapezoid, the diagonals are congruent, so we only need to find the length of one diagonal. Let's call the diagonal d.

Using the Pythagorean theorem, we can find the length of the diagonal:
d = √(h^2 + (b2 - b1)^2)

Where h is the height of the trapezoid, b1 is the shorter base, and b2 is the longer base.

In this case, we know that the shorter base (b1) is 7 feet long and the longer base (b2) is 22 feet long.

To find the height, we can use the fact that the sum of the base angles (angle A and angle B) is 140°. Since the opposite angles are congruent in an isosceles trapezoid, we can conclude that each base angle is (180° - 140°) / 2 = 20°.

Now we can use trigonometry to find the height. Drawing a right triangle within the trapezoid, we have:
tan(20°) = height / (b2 - b1)
tan(20°) = height / (22 - 7)
height = tan(20°) * 15

Substituting the values into the formula for the diagonal length:
d = √(height^2 + (b2 - b1)^2)
d = √((tan(20°) * 15)^2 + (22 - 7)^2)

Using a calculator, evaluate the expression to find the length of the diagonal.

b. Find the length of the shorter base:

In an isosceles trapezoid, the bases are parallel, and the opposite sides are equal. So the length of the shorter base (b1) is also 7 feet.

Therefore, the length of the shorter base is 7 feet.

Round the answers to the nearest hundredth as requested.