Question

which pair of triangles have equivalent areas shaded? choose 1 answer: a two triangles with shaded regions = two triangles with shaded regions b a small triangle with shaded region = a larger triangle with shaded region c a small triangle with shaded region = a larger triangle with shaded region

Explanation:

Step1: Recall the formula for the area of a triangle

The area of a triangle is \(A=\frac{1}{2}bh\) (where \(b\) is the base and \(h\) is the height). When a triangle is divided into smaller non - overlapping triangles with the same height (or same base), the ratio of their areas is equal to the ratio of their bases (or heights).

Step2: Analyze Option A

In the first pair of triangles in Option A:
Let's assume the large triangle has a certain height \(h\). The smaller sub - triangles (shaded and un - shaded) in each of the two larger triangles have the same height.
If we consider the first triangle (left - hand side of the equal sign in Option A), assume the base of the large triangle is \(b\). The two shaded sub - triangles have bases \(b_1\) and \(b_2\) such that \(b = b_1 + b_2\). In the second triangle (right - hand side of the equal sign in Option A), using the property that triangles with the same height, the sum of the areas of the shaded sub - triangles (since they have the same height as the sub - triangles in the first triangle) is the same.
The area of a sub - triangle is \(A_i=\frac{1}{2}b_ih\). If we have two sub - triangles in the left - hand side triangle \(A_{left}=\frac{1}{2}b_1h+\frac{1}{2}b_2h=\frac{1}{2}(b_1 + b_2)h\), and for the right - hand side triangle (with the same \(h\) and \(b_1 + b_2\) as the sum of the bases of the shaded sub - triangles) \(A_{right}=\frac{1}{2}(b_1 + b_2)h\)

Step3: Analyze Option B

For Option B, assume the height of the smaller triangle is \(h_1\) and the height of the larger triangle is \(h_2\) (\(h_2>h_1\)). Using the formula \(A=\frac{1}{2}bh\), even if the sum of the bases of the shaded sub - triangles in the larger triangle is the same as the base of the smaller triangle, since \(h\) is different (\(A_1=\frac{1}{2}b h_1\) and \(A_2=\frac{1}{2}b h_2\) with \(h_1
eq h_2\)), the areas are not equal.

Step4: Analyze Option C

For Option C, assume the height of the smaller triangle is \(h_1\) and the height of the larger triangle is \(h_2\) (\(h_2 > h_1\)). Using the formula \(A=\frac{1}{2}bh\), if we assume the base of the smaller triangle is \(b_1\) and the base of the larger triangle is \(b_2\) (even if the number of sub - triangles is the same), since \(A_1=\frac{1}{2}b_1h_1\) and \(A_2=\frac{1}{2}b_2h_2\) and \(b_1h_1
eq b_2h_2\) (because of the size difference), the areas are not equal.

Answer:

A.