I'm currently working on a geometry problem that involves finding the length of the two sides of a triangle, given a base length of 70 mm. The problem is presented in two statements, and I'm having difficulty understanding why one statement alone is sufficient while the other is not.
Statement (1) says that the height of the triangle divides the base into two equal parts. Statement (2) says that the angles in the triangle are equal. The question asks which of the statements is sufficient to solve the problem.
From what I understand, if statement (2) is true, then the triangle must be equilateral, which means that all sides have the same length of 70 mm. Therefore, statement (2) alone is sufficient to solve the problem.
However, I'm not sure how statement (1) on its own doesn't provide enough information. I know that statement (1) means that the triangle is isosceles, which means that two of the sides are equal. But how can we be sure that the triangle is isosceles from statement (1) alone?
I would really appreciate any help or guidance on this problem. Thank you in advance!
Statement 1 on its own, merely establishes that the triangle is isosceles. This means, that while you know that two of the angles are equal, you don't know what these two equal angles are, and you therefore, don't know what the other angle opposite the base is.
So, all that you know is that you have an isosceles triangle, with a specific base length given. From this (incomplete) information, the triangle is not determined, and therefore can not be solved.
To see this graphically, construct the base, and construct the perpendicular bisector to the base. You can choose any other point on the perpendicular bisector to be the third vertex of the triangle, and still satisfy the incomplete constraints. This graphically illustrates that the triangle is undetermined, and therefore can not be solved.