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  3. Which of the following statements individually provide(s)sufficient additional information to determine the area of triangle ABC

Which of the following statements individually provide(s)sufficient additional information to determine the area of triangle ABC

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问题
Which of the following statements individually provide(s)sufficient additional information to determine the area of triangle ABC above? Indicate all such statements.
选项 A、DBC is an equilateral triangle.
B、ABD is an isosceles triangle.
C、The length of BC is equal to the length of AD.
D、The length of BC is 10.
E、The length of AD is 10.
答案A，D
解析 From the figure you 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. However, because the figure is not necessarily drawn to scale, you don’t know the lengths of AD, DC, and BC. In particular, you don’t know where D is on AC.
    The area of a triangle is 1/2(base)(height). Thus, the area of right triangle ABC is equal to 1/2 of the length of AB times the length of BC. You already know that the length of AB is 10. Any additional information that would allow you to calculate the length of BC would be sufficient to find the area of triangle ABC. You 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 60° angle; and therefore, triangle ABC is a 30°-60°-90° 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. You have no way of 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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