Answer:
a) i) 1 : 3
a) ii) 3 : 5
a) iii) 5 : 7
b) (2n - 1) : (2n + 1)
Step-by-step explanation:
Let b be the base of the larger triangle.
Let h be the height of the larger triangle.
Question a)i)
The triangle is divided into 2 equal-width sections. The rightmost section forms a smaller, similar shaded triangle. The base and height of the shaded triangle are each half those of the larger triangle.
The area of a triangle is half the product of its base and height. So, the area of the shaded triangle is:
[tex]\textsf{Shaded area}=\dfrac12\left(\dfrac12b\cdot\dfrac12h\right)=\dfrac18bh[/tex]
The unshaded area can be calculated by subtracting the shaded area from the area of the larger triangle:
[tex]\textsf{Unshaded area}=\dfrac12bh-\dfrac18bh=\dfrac38bh[/tex]
So, the ratio of the shaded area to the unshaded area is:
[tex]\dfrac18bh:\dfrac38bh=\boxed{1:3}[/tex]
[tex]\dotfill[/tex]
Question a)ii)
The triangle is divided into 4 equal-width sections. The rightmost section forms a smaller, similar shaded triangle. The base and height of the shaded triangle are each a quarter of those of the larger triangle.
The area of the smallest shaded triangle is:
[tex]\textsf{Shaded area 1}=\dfrac12\left(\dfrac14b\cdot\dfrac14h\right)=\dfrac{1}{32}bh[/tex]
The unshaded area to the left of the smallest shaded triangle is the difference between the area of the smallest shaded triangle and a similar triangle with half the base and height of the larger triangle.
[tex]\textsf{Unshaded area 1}=\dfrac12\left(\dfrac12b\cdot \dfrac12h\right)-\dfrac{1}{32}bh=\dfrac{3}{32}bh[/tex]
The shaded area to the left of unshaded area 1 is found by subtracting the area of a triangle with half the base and height of the larger triangle from the area of a triangle with 3/4 the base and height of the larger triangle.
[tex]\textsf{Shaded area 2}=\dfra12\left(\dfrac34b\cdot\dfrac34h\right)-\dfrac12\left(\dfrac12b\cdot \dfrac12h\right)=\dfrac{5}{32}bh[/tex]
The unshaded area to the left of shaded area 2 is found by subtracting the area of a triangle with 3/4 the base and height of the larger triangle from the area of the larger triangle:
[tex]\textsf{Unshaded area 2}=\dfrac12bh-\dfrac12\left(\dfrac34b\cdot\dfrac34h\right)=\dfrac{7}{32}bh[/tex]
So, the total shaded area is:
[tex]\textsf{Total shaded area}=\dfrac{1}{32}bh+\dfrac{5}{32}bh=\dfrac{3}{16}bh[/tex]
The total unshaded area is:
[tex]\textsf{Total unshaded area}=\dfrac{3}{32}bh+\dfrac{7}{32}bh=\dfrac{5}{16}bh[/tex]
So, the ratio of the shaded area to the unshaded area is:
[tex]\dfrac{3}{16}bh:\dfrac{5}{16}bh=\boxed{3:5}[/tex]
[tex]\dotfill[/tex]
Question a)iii)
The triangle is divided into 6 equal-width sections. The rightmost section forms a smaller, similar shaded triangle. The base and height of the shaded triangle are each a sixth of those of the larger triangle.
(Please see attachments for full explanation of this part).
[tex]\textsf{Shaded area 1}=\dfrac12\left(\dfrac16b\cdot\dfrac16h\right)=\dfrac{1}{72}bh[/tex]
[tex]\textsf{Unshaded area 1}=\dfrac12\left(\dfrac26b\cdot\dfrac26h\right)-\dfrac{1}{72}bh=\dfrac{1}{24}bh[/tex]
[tex]\textsf{Shaded area 2}=\dfrac12\left(\dfrac36b\cdot\dfrac36h\right)-\dfrac12\left(\dfrac26b\cdot\dfrac26h\right)=\dfrac{5}{72}bh[/tex]
[tex]\textsf{Unshaded area 2}=\dfrac12\left(\dfrac46b\cdot \dfrac46h\right)-\dfrac12\left(\dfrac36b\cdot\dfrac36h\right)=\dfrac{7}{72}bh[/tex]
[tex]\textsf{Shaded area 3}=\dfrac12\left(\dfrac56b\cdot\dfrac56h\right)-\dfrac12\left(\dfrac46b\cdot\dfrac46h\right)=\dfrac18bh[/tex]
[tex]\textsf{Unshaded area 3}=\dfrac12bh-\dfrac12\left(\dfrac56b\cdot\dfrac56h\right)=\dfrac{11}{72}bh[/tex]
So, the total shaded area is:
[tex]\textsf{Total shaded area}=\dfrac{1}{72}bh+\dfrac{5}{72}bh+\dfrac18bh=\dfrac{5}{24}bh[/tex]
The total unshaded area is:
[tex]\textsf{Total unshaded area}=\dfrac{1}{24}bh+\dfrac{7}{72}bh+\dfrac{11}{72}bh=\dfrac{7}{24}bh[/tex]
So, the ratio of the shaded area to the unshaded area is:
[tex]\dfrac{5}{24}bh:\dfrac{7}{24}bh=\boxed{5:7}[/tex]
[tex]\dotfill[/tex]
Question b)
The ratios of the shaded area to the unshaded area when the triangle is split into 2n sections is:
[tex]n=1\implies 1:3[/tex]
[tex]n=2\implies 3:5[/tex]
[tex]n=3\implies 5:7[/tex]
The pattern for the ratios is that the first number is always an odd number, and the second number is always 2 greater than the first.
So, the general rule for this ratio is:
[tex]\boxed{(2n - 1):(2n + 1)}[/tex]
