Solution

From the figure we know that ABC is a right triangle with its right angle at vertex B. You also know that point D is on the hypotenuse AC. You are given that the length of AB is \(10\sqrt{3}\). However, because the figure is not necessarily drawn to scale, we don’t know the lengths of AD, DC, and BC. In particular, we don’t know where D is on AC.

The area of a triangle is \(\frac{1}{2}(base)(height)\). Thus, the area of right triangle ABC is equal to 1 of the length of AB times the length of BC. We already know that the length of AB is 103. Any additional information that would allow us to calculate the length of BC would be sufficient to find the area of triangle ABC.

We need to consider each of the five statements individually, as follows.

Statement A: DBC is an equilateral triangle. This statement implies that angle DCB is a 60degree angle; and therefore, triangle ABC is a 30−60−90 degree triangle. Thus the length of BC can be determined, and this statement provides sufficient additional information to determine the area of triangle ABC.

Statement B: ABD is an isosceles triangle. There is more than one way in which triangle ABD can be isosceles. Below are two redrawn figures showing triangle ABD as isosceles. In the figure on the left, the length of AD is equal to the length of DB; and in the figure on the right, the length of AB is equal to the length of AD.




Either of the figures could have been drawn with the length of BC even longer. So, statement B does not provide sufficient additional information to
determine the area of triangle ABC.

Statement C: The length of BC is equal to the length of AD. We have no wayof finding the length of AD without making other assumptions, so statement C does not provide sufficient additional information to determine the area of triangle ABC.

Statement D: The length of BC is 10. The length of BC is known, so the area of triangle ABC can be found. Statement D provides sufficient additional information to determine the area of triangle ABC.

Statement E: The length of AD is 10. The relationship between AD and BC is not known, so statement E does not provide sufficient additional information to determine the area of triangle ABC.

Statements A and D individually provide sufficient additional information to determine the area of triangle ABC. Therefore, the correct answer consists of Choices A and D.







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Can u explain exactly how to use Statement A to find the length of BC?

Statement A states that the Triangle "DBC is an equilateral triangle" which means all angles within the DBC Triangle will have 60. Now since the question gives you the angle B of the Triangle ABC (90) we can figure out the angle A to be 30. (180-90[the right angle] - 60 [Angle C])
Now we can use the 30-60-90 ratio to find the length of BC to find the are of the triangle ABC.

Look at the info given..
We know one angle that is 90° and one side...
what do we require other than this to make a unique triangle..
It is either one one side or one more angle..

A gives us another angle that is 60°, so triangle is unique and we can find the area.
D gives us length of a side, again unique triangle.

Rest all do not help in finding either side or angle and thus insufficient..

Answer A and D
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A) DBC is an equilateral triangle.
This means we can add the following to our diagram.



At this point, we see we have a special 30-60-90 right triangle, which is 10 TIMES larger than the base 30-60-90 right triangle

At this point, we can calculate the area of triangle ABC

Area = (base)(height)/2
= (10)(10√3)/2
= 50√3

So, statement A is sufficient to determine the area of triangle ABC
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B) ABD is an isosceles triangle.
Consider these two triangles that satisfy the give conditions

Clearly, the area of triangle ABC in the TOP diagram is different from is the area of triangle ABC in the BOTTOM diagram

So, statement B is NOT sufficient to determine the area of triangle ABC
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Consider these two triangles that satisfy the give conditions
C) The length of BC is equal to the length of AD.

Clearly, the area of triangle ABC in the TOP diagram is different from is the area of triangle ABC in the BOTTOM diagram

So, statement C is NOT sufficient to determine the area of triangle ABC

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D) The length of BC is 10.

This provides enough information to calculate the area of triangle ABC

Area = (base)(height)/2
= (10)(10√3)/2
= 50√3

So, statement D is sufficient to determine the area of triangle ABC


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Consider these two triangles that satisfy the give conditions
E) The length of AD is 10.

Clearly, the area of triangle ABC in the TOP diagram is different from is the area of triangle ABC in the BOTTOM diagram

So, statement E is NOT sufficient to determine the area of triangle ABC


Answer: A & D

Cheers,
Brent
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