Answer:
x = 8
Step-by-step explanation:
Similar shapes are dilations of each other using a scale factor.
As the area of both figures has been given, the scale factor that maps the green figure to the yellow figure can be calculated:
[tex]\implies \textsf{Scale factor (area)} = \dfrac{\textsf{area of yellow figure}}{\textsf{area of green figure}} = \sf \dfrac{352}{22} = 16[/tex]
Area is measured in square units. Therefore, to find the scale factor for length, square root the scale factor for area:
[tex]\begin{aligned}\implies \textsf{Scale factor (length)} & = \sf \sqrt{\textsf{Scale factor for area}} \\ & =\sf \sqrt{16}\\ & = \sf 4 \end{aligned}[/tex]
Use the found scale factor for length to find the perimeter of the yellow figure:
[tex]\begin{aligned} \sf \implies \textsf{Perimeter (yellow fig)} & = \sf \textsf{perimeter (green fig)} \times scale\:factor \\ & = \sf 26 \times 4 \\ & = \sf 104\: mm \end{aligned}[/tex]
Find the missing horizontal lengths of the yellow figure by using the corresponding length of the green figure:
⇒ 4 mm × 4 = 16 mm
⇒ 24 mm - 16 mm = 8 mm
Let the height of the yellow figure be y.
⇒ 24 + 8 + 16 + y + x + (y - x) = perimeter
⇒ 48 + 2y = 104
⇒ 2y = 56
⇒ y = 28 mm
Divide the yellow figure into two rectangles (see attachment).
- Area of left rectangle = y × 8 = 28 × 8 = 224 mm²
- Area of right rectangle = 16x mm²
To find x, sum the areas of the rectangles and equate to 352:
⇒ 224 + 16x = total area
⇒ 224 + 16x = 352
⇒ 16x = 128
⇒ x = 8
