Question

1. understand given kite abcd and trapezoid wacz, complete a two - column proof to show that wacz is an isosceles trapezoid.

statement | reason

1. | 1. given
2. $overline{ad} \cong \overline{cd}$ | 2.
3. | 3. diagonals of a kite are $\perp$.
4. $overline{ed} \cong \overline{ed}$ | 4.
5. | 5. hl thm.
6. $\angle dae \cong \angle dce$ | 6.
7. | 7. alt. int. $\angle$s thm.
8. | 8. transitive prop. of $\cong$
9. $\triangle adw \cong \triangle cdz$ | 9.
10. $\angle awd \cong \angle czd$ | 10.
11. wacz is an isosceles trapezoid. | 11.
12. apply in bhutanese architecture, the faces of buildings are trapezoids. if the row of decorative brick shown is 28.5 feet wide, what is the width of the base of the building?

Explanation:

Step1: Identify the trapezoid type

The building's face is an isosceles trapezoid (symmetric, as seen from the diagram with equal non - parallel sides of 16 ft each). In an isosceles trapezoid, the non - parallel sides (legs) are equal, and the segments on the base formed by the mid - segment (the decorative brick row) can be calculated using the formula related to the mid - segment of a trapezoid. The mid - segment \(m\) of a trapezoid is given by \(m=\frac{a + b}{2}\), where \(a\) and \(b\) are the lengths of the two parallel sides (the top and the base of the trapezoid). Here, the mid - segment (decorative brick row) length \(m = 28.5\) ft, and the top side \(a=25\) ft. We need to find the base \(b\).

Step2: Rearrange the mid - segment formula

From \(m=\frac{a + b}{2}\), we can multiply both sides by 2 to get \(2m=a + b\). Then, we can solve for \(b\) by subtracting \(a\) from both sides: \(b = 2m-a\).

Step3: Substitute the values

Substitute \(m = 28.5\) ft and \(a = 25\) ft into the formula. \(b=2\times28.5 - 25\). First, calculate \(2\times28.5=57\) ft. Then, \(b = 57-25=32\) ft.

Answer:

The width of the base of the building is \(\boldsymbol{32}\) feet.