Figure It Out: Area & Patterns

Area of Dashed Region

The figure shows a rectangle with a corner removed (L-shape). Let outer dimensions be $p$ and $s+r$. Inner cutout dimensions are $p-r$ and $s$.

Method 1 (Subtraction): Total Area - Cutout Area $p(s+r) - s(p-r)$ $= ps + pr - sp + sr$ $= pr + sr = r(p+s)$

Method 2 (Addition of Rectangles): Split into bottom rectangle ($s \times r$) and side rectangle ($p \times r$). $sr + pr = r(s+p)$

Calculation: $p=6, r=3.5, s=9$. Area $= 3.5(6 + 9) = 3.5 \times 15 = 52.5$.

Figure It Out 2 (Expansions)

(i) $(p-1)(p+11)$ $= p^2 + 11p - p - 11 = p^2 + 10p - 11$

(ii) $(3a-9b)(3a+9b)$ Difference of squares: $(3a)^2 - (9b)^2 = 9a^2 - 81b^2$.

(iii) $-(2y+5)(3y+4)$ $-[6y^2 + 8y + 15y + 20] = -(6y^2 + 23y + 20)$

(iv) $(6x+5y)^2$ $36x^2 + 60xy + 25y^2$

(v) $(2x - \frac{1}{2})^2$ $4x^2 - 2(2x)(\frac{1}{2}) + \frac{1}{4} = 4x^2 - 2x + 0.25$