MHB Find the area of the red region

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The discussion focuses on calculating the area of a red region within a parallelogram, where the areas of adjacent green regions are provided as 8 unit², 10 unit², 72 unit², and 79 unit². Participants analyze the relationships between these areas to derive the area of the red region. The total area of the parallelogram is implied to be the sum of the green regions plus the red region. By applying geometric principles and area calculations, the participants aim to find the value of the red region. The conclusion emphasizes the importance of understanding area relationships in geometric figures.
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The diagram below (which is not drawn to scale) shows a parallelogram. The area of the green regions are 8 unit² , 10 unit² , 72 unit² and 79 unit² respectively. Find the area of the red region.

[TIKZ]
\coordinate (A) at (0,0);
\coordinate (B) at (8,0);
\coordinate (C) at (12,0);
\coordinate (D) at (12.75,3);
\coordinate (E) at (14,8);
\coordinate (F) at (6,8);
\coordinate (G) at (2,8);
\coordinate[label=above: \huge 79] (P) at (4.5,3);
\coordinate[label=above: \huge 72] (Q) at (9.5,6);
\coordinate[label=above: \large 8] (R) at (8.5,1.2);
\coordinate[label=above: \large 10] (S) at (11,2.9);
\draw (A) -- (C)-- (E) -- (G) -- (A);
\draw (A) -- (F);
\draw (A) -- (D);
\draw (B) -- (F);
\draw (B) -- (E);
\draw (D) -- (G);
\draw[fill=teal] (0,0) -- (7.564,1.745) -- (6.513,5.949) -- (4.983,6.644);
\draw[fill=teal] (6,8) -- (6.508,5.931) -- (10.93,3.85) -- (14,8);
\draw[fill=teal] (8,0) -- (7.564,1.76) -- (9.674,2.25);
\draw[fill=teal] (10.92,3.89) -- (12.75,3) -- (9.674,2.25);
\draw[fill=teal] (4.5,3) node[text=pink] {\huge 79};
\draw[fill=teal] (9.5,6) node[text=pink] {\huge 72};
\draw[fill=teal] (8.5,1.2) node[text=pink] {\large 8};
\draw[fill=teal] (11,2.9) node[text=pink] {\large 10};
\draw[fill=magenta] (2,8) -- (4.983,6.644) -- (6,8);
[/TIKZ]
 
Last edited:
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[TIKZ]
\coordinate (A) at (0,0);
\coordinate (B) at (8,0);
\coordinate (C) at (12,0);
\coordinate (D) at (12.75,3);
\coordinate (E) at (14,8);
\coordinate (F) at (6,8);
\coordinate (G) at (2,8);
\coordinate[label=above: \huge 79] (P) at (4.5,3);
\coordinate[label=above: \huge 72] (Q) at (9.5,6);
\coordinate[label=above: \large 8] (R) at (8.5,1.2);
\coordinate[label=above: \large 10] (S) at (11,2.9);
\coordinate[label=above: \huge A] (M) at (2.5,5);
\coordinate[label=above: \huge B] (N) at (6,0.48);
\coordinate[label=above: \huge C] (Z) at (5.8,6.8);
\coordinate[label=above: \huge D] (J) at (8.5,3.2);
\coordinate[label=above: \huge E] (K) at (10.5,0.9);
\coordinate[label=above: \huge F] (I) at (12.2,4.2);
\draw (A) -- (C)-- (E) -- (G) -- (A);
\draw (A) -- (F);
\draw (A) -- (D);
\draw (B) -- (F);
\draw (B) -- (E);
\draw (D) -- (G);
\draw[fill=teal] (0,0) -- (7.564,1.745) -- (6.513,5.949) -- (4.983,6.644);
\draw[fill=teal] (6,8) -- (6.508,5.931) -- (10.93,3.85) -- (14,8);
\draw[fill=teal] (8,0) -- (7.564,1.76) -- (9.674,2.25);
\draw[fill=teal] (10.92,3.89) -- (12.75,3) -- (9.674,2.25);
\draw[fill=teal] (4.5,3) node[text=pink] {\huge 79};
\draw[fill=teal] (9.5,6) node[text=pink] {\huge 72};
\draw[fill=teal] (8.5,1.2) node[text=pink] {\large 8};
\draw[fill=teal] (11,2.9) node[text=pink] {\large 10};
\draw[fill=magenta] (2,8) -- (4.983,6.644) -- (6,8);
[/TIKZ]

If we look at the parallelogram in such a way that the horizontal sides are the base, then we have

$\normalsize \text{Area of B}+\text{Area of C}+79+\text{Area of E}+\text{Area of F}+10=\text{Area of A}+\text{Area of red region}+72+8+\text{Area of D}$

If we look at the parallelogram in such a way that the slanted sides are the base, then we have

$\normalsize \text{Area of B}+\text{Area of E}+8+\text{Area of F}+\text{Area of C}+72+\text{Area of red region}=\text{Area of A}+79+10+\text{Area of D}$

Subtracting the below from the above we get

$ 9-\text{Area of red region}=-9+\text{Area of red region}\\ \\ \therefore \text{Area of red region}=9$
 
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