The ratio of the bases of two triangles is x : y and the ratio of their areas is a : b. What is the ratio of their corresponding altitudes?

Introduction:
This is a conceptual question about the relationship between the areas, bases, and altitudes (heights) of two triangles. The problem gives the ratio of the bases and the ratio of the areas and then asks for the ratio of the corresponding altitudes. This type of question tests whether students are comfortable manipulating proportional relationships and applying the area formula for a triangle, which involves both base and height.

Given Data / Assumptions:

• We have two triangles, say triangle 1 and triangle 2. • Ratio of their bases: base1 : base2 = x : y. • Ratio of their areas: area1 : area2 = a : b. span style="display:block;">• Let h1 and h2 be the corresponding altitudes (heights) for bases base1 and base2 respectively. • We need to find h1 : h2 in terms of a, b, x, and y.

Concept / Approach:
The area of a triangle is given by area = (1/2) * base * height. For each triangle, area1 = (1/2) * base1 * h1 and area2 = (1/2) * base2 * h2. If we divide area1 by area2, the factor (1/2) cancels out, and the ratio of areas becomes a ratio involving base1, base2, h1, and h2. Using the given ratios area1 : area2 = a : b and base1 : base2 = x : y, we can manipulate this relationship to solve for the ratio h1 : h2.

Step-by-Step Solution:
Step 1: Write expressions for areas of the two triangles. Area1 = (1/2) * base1 * h1. Area2 = (1/2) * base2 * h2. Step 2: Consider the ratio of the areas. Area1 / Area2 = [(1/2) * base1 * h1] / [(1/2) * base2 * h2] = (base1 * h1) / (base2 * h2). Step 3: Use the given area ratio. Area1 / Area2 = a / b. So, (base1 * h1) / (base2 * h2) = a / b. Step 4: Insert the base ratio base1 / base2 = x / y. (base1 / base2) * (h1 / h2) = a / b. So, (x / y) * (h1 / h2) = a / b. Step 5: Solve for h1 / h2. h1 / h2 = (a / b) * (y / x) = (a * y) / (b * x). Thus, h1 : h2 = ay : bx.

Verification / Alternative check:
To verify, take a concrete example. Let x = 2, y = 3, a = 4, and b = 6. Then base1 : base2 = 2 : 3 and area1 : area2 = 4 : 6 = 2 : 3. From the formula, h1 : h2 should be (a * y) : (b * x) = 4 * 3 : 6 * 2 = 12 : 12 = 1 : 1. If base1 = 2 units and base2 = 3 units, and h1 = h2, the area ratio is (1/2 * 2 * h1) : (1/2 * 3 * h1) = 2h1 : 3h1 = 2 : 3, consistent with the given ratio. This supports our derived result.

Why Other Options Are Wrong:
Option a : b: This ignores the effect of the base ratio and would be correct only if the bases were equal. Option x : y: This assumes that area ratio has no effect on heights, which contradicts the formula for area. Option bx : ay: This is simply the inverse of the correct ratio and would be obtained by mistakenly rearranging the equation. Option x * a : y * b: While superficially similar, this does not reduce to the derived ratio and misplaces which factors multiply together.

Common Pitfalls:
Students sometimes incorrectly cancel the base and height terms or forget that area of triangle depends on both base and height, not just one dimension. Another common error is to directly equate area ratio with height ratio, overlooking the base ratio. Careful algebraic manipulation of the area formula and stepwise reasoning prevents such mistakes.

Final Answer:
The ratio of the corresponding altitudes is ay : bx.