Math Help Please?

Question

The area of a triangle is divided into 3 equal parts by line segments parallel to one side. If the length of that side is 18 cm, find the length of the longest line segment inside the triangle, in cm.

Answer 1

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The triangle is ABC with AB = 18cm

. .C

. P---Q

.R------S

A. . . . . .B

Lines PQ and RS split ABC into 3 equal areas, PQC, RSQP and ABSR.

We want length RS.

Note ΔABC is similar to ΔRSC.  And the area of ΔRSC is 2/3 the area of ΔABC.

Since areas of similar triangles are proportional to lengths of their corresponding sides:

RS²/AB² = 2/3

RS = AB√(2/3) = 18√(2/3) = 6√6 cm = 14.7cm approx.

Answer 2

Hint2 similar area A1 , A2 , sides x1 , x2

A1/A2=(x1/x2)^2

(---> the length of the longest line segment inside the triangle is 6√6= ...)

Answer 3

Top triangle has less width so longer sides to compensate and equal other areas.

Let x be the side length of the top triangle.

Areas proportional to square of side length, so,

x^2/18^2 = 1/3

x = 6√(3) ~ 10.39 cm