Triangle ABC has an obtuse angle at B, base BC has length equal to 30 and height equal to 24. (This height is taken with respect to base BC). D is a point on the line segment BC and E is a point on AC such that DE∥AB. F is a point on AB such that FD∥AC. As D varies within line segment BC, what is the maximum value of [DEF]?
Shame on you Keshav!!! Cheating on Brilliant!!! This site is meant to be a platform to practice your own skills, not to copy paste the questions and get free answers and then get incentives without effort. So either play fair and be honest or leave this site. People like you are shame to the Brilliant community. And to the others, please give the answer to this problem after Monday 10/6/2013, so that this cheat doesn't get the opportunity to cheat.
Thanks for noticing, Shame. As such we have started tracking keshav's and mathlover's account( i.e. we are searching which accounts got these problems, and we are searching which accounts entered exactly the answers posted here, even if they are wrong at moreorless the same time or date). Currently we have pinpointed about five possibilities for keshav's account. A few more posts and he will be ours. Thanks for your cooperation, Shame.
-Calvin
Brilliant Maths Challenge Master
To determine the maximum value of the area of triangle DEF ([DEF]), we need to consider the relationship between the areas of triangles ABC and DEF.
Step 1: Calculate the area of triangle ABC.
The area of triangle ABC can be calculated using the formula: Area = 1/2 * base * height.
In this case, the base BC has a length of 30 and the height relative to BC is 24. Therefore, the area of triangle ABC is 1/2 * 30 * 24 = 360.
Step 2: Determine the relationship between the areas of triangles ABC and DEF.
Since DE∥AB and FD∥AC, we know that triangles ABC and DEF are similar. This means that the ratio of their areas is equal to the square of the ratio of their corresponding sides.
Step 3: Determine the ratio of the sides of triangles DEF and ABC.
Let's label the lengths of the sides of triangle DEF as a, b, and c (in correspondence with sides DE, EF, and FD, respectively). Similarly, we label the corresponding sides of triangle ABC as A, B, and C (in correspondence with sides BC, CA, and AB, respectively).
From the similarity of the triangles, we have the following ratios:
a/BC = b/CA = c/AB. (1)
To establish a relationship between the sides of triangle DEF and triangle ABC, we can use ratios of similar triangles.
Note that the height of triangle DEF (with respect to base EF) is the same as the height of triangle ABC (with respect to base BC). Therefore, we have:
a/EF = A/BC. (2)
Furthermore, from the given information, we know that FD∥AC. This means that triangles ABC and DEF are similar, so their corresponding sides are in proportion:
b/EF = B/CA. (3)
Step 4: Calculate the maximum value of [DEF].
To find the maximum value of [DEF], we need to maximize the area of the triangle subject to the constraints of the problem. Since triangles ABC and DEF are similar, maximizing the area of triangle DEF would be equivalent to maximizing the area ratio between the two triangles.
Using the ratios from steps 2 and 3, we can express the area ratio of triangles DEF and ABC as follows:
[DEF]/ABC = (a*b*c)/(BC*CA*AB).
To maximize [DEF], we can maximize the area ratio. From the given information, we have the following conditions:
- BC = 30 (given)
- AB and CA are fixed as they represent the sides of triangle ABC.
- For the ratio (a*b*c), we want to find the maximum values for a and b.
Based on equation (1), we have the condition a/30 = b/CA. Therefore, as b/CA increases, a must also increase to satisfy the equation. This means that to maximize (a*b*c), we consider the largest possible value for b (denoted as b_max) and the largest possible value for a (denoted as a_max).
Step 5: Determine the maximum value of [DEF].
To find the maximum value of [DEF], we substitute the maximum values of a and b into the equation:
[DEF]/ABC = (a_max * b_max * c)/(30 * CA * AB).
However, since we are only interested in maximizing [DEF], we can disregard the constant terms 30, CA, and AB.
Therefore, the maximum value of [DEF] would be (a_max * b_max * c).
Note that the values of a_max and b_max will depend on the specific values of AB, BC, and CA, which are not provided in the given question. To find the exact maximum value, the specific lengths of these sides would need to be known.