In the sequence below, each shaded triangle, after the first is 1/4 the area of the preceding shaded triangle. What is the ratio for the shaded area of the 5th triangle?
We have no access to your figure.
To find the ratio for the shaded area of the 5th triangle, we need to understand the pattern in the sequence of triangles. Let's break it down step by step:
1st triangle:
Assume the area of the shaded part is A.
The first triangle is not mentioned, so we don't know the size of the shaded area.
2nd triangle:
The shaded area of the 2nd triangle is 1/4 of the preceding triangle's shaded area.
This means the area of the shaded part in the 2nd triangle is (1/4)A, which is A/4.
3rd triangle:
The shaded area of the 3rd triangle is 1/4 of the 2nd triangle's shaded area.
So, the area of the shaded part in the 3rd triangle is (1/4)(A/4) = A/16.
4th triangle:
Similar to the previous steps, the shaded area of the 4th triangle is 1/4 of the 3rd triangle's shaded area.
Therefore, the area of the shaded part in the 4th triangle is (1/4)(A/16) = A/64.
5th triangle:
Again, the shaded area of the 5th triangle is 1/4 of the 4th triangle's shaded area.
So, the area of the shaded part in the 5th triangle is (1/4)(A/64) = A/256.
Therefore, the ratio for the shaded area of the 5th triangle is A/256.