Question

a frame shop charges according to both the amount of framing needed to surround the picture and the amount of glass needed to cover the picture.
a. find the area and perimeter of the trapezoid - shaped framed picture to the right.
b. identify whether the frame has to do with perimeter or area and the same with the glass.
a. the area of the picture is (square) sq in. and the perimeter is (square) in.

Explanation:

Step1: Recall trapezoid area formula

The area formula of a trapezoid is $A=\frac{(a + b)h}{2}$, where $a$ and $b$ are the lengths of the parallel - sides and $h$ is the height. Here, $a = 19$ in, $b = 51$ in, and $h = 12$ in.
$A=\frac{(19 + 51)\times12}{2}$

Step2: Calculate the area

First, add the lengths of the parallel - sides: $19+51 = 70$ in. Then multiply by the height: $70\times12=840$ in². Finally, divide by 2: $\frac{840}{2}=420$ in².

Step3: Recall trapezoid perimeter formula

The perimeter $P$ of a trapezoid is the sum of the lengths of all its sides. Here, the side - lengths are $19$ in, $20$ in, $51$ in, and $20$ in.
$P=19 + 20+51 + 20$

Step4: Calculate the perimeter

$19+20 = 39$ in, $39+51 = 90$ in, $90+20 = 110$ in.

Step5: Analyze relationship for part b

The amount of framing needed is related to the perimeter of the trapezoid (the length around the outside), and the amount of glass needed is related to the area of the trapezoid (the surface to be covered). The perimeter and area are different measures and have different units (length for perimeter and area for area), so they are not the same.

Answer:

a. The area of the picture is $420$ sq in. and the perimeter is $110$ in.
b. The frame is related to the perimeter and the glass is related to the area, and they are not the same.