swerve wrote:Patricia builds two triangles, each with 30 feet of wood. The first triangle ABC is built to maximize the length of the base side. The second triangle DEF is built to maximize the area of the triangle. What is the ratio of the length of the base of triangle ABC to the length of the base of triangle DEF? The lengths of all line segments are integers.

A. 1:1
B. 6:5
C. 5:4
D. 7:5
E. 2:1

The OA is D

Source: Veritas Prep

So, we have two triangles with the perimeter 30 feet given that the sides of the triangles are integers.

1. Triangle-1 has the maximum possible base:

The maximum possible base would be a little less than half the perimeter of the triangle.

Thus, the maximum possible base < 1/2 of 30 (= 15)

The maximum possible base or base of ∆ABC = 14 since it is given that the sides of the triangles are integers.

1. Triangle-2 has the maximum possible area:

For the given perimeter 30, the maximum possible area would be for a triangle that has the same sides; thus, it is an equilateral triangle.

Thus, side of an equilateral triangle of ∆DFF = 30/3 = 10 feet

Ratio of the length of the base of triangle ABC to the length of the base of triangle DEF = 14/10 = 7 : 5.

The correct answer: D

Hope this helps!

-Jay
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swerve wrote:Patricia builds two triangles, each with 30 feet of wood. The first triangle ABC is built to maximize the length of the base side. The second triangle DEF is built to maximize the area of the triangle. What is the ratio of the length of the base of triangle ABC to the length of the base of triangle DEF? The lengths of all line segments are integers.

A. 1:1
B. 6:5
C. 5:4
D. 7:5
E. 2:1

The OA is D

Source: Veritas Prep

Since triangle ABC maximizes the length of the base, and a side of the triangle must be less than the sum of the other two sides, then the maximum length of the base will be 14, which is 1 less than half of the perimeter.

The triangle with a fixed perimeter that maximizes the area is an equilateral triangle since, if a polygon is given a fixed perimeter, the polygon with the largest area must be regular (note: a "regular" triangle is an equilateral triangle). So a side (or base) of triangle DEF = 10.

Therefore, the ratio of the the two bases = 14/10 = 7/5.

Answer: D