AMC 10/12 Student Practice Questions
1 answer:
Answer:
Step-by-step explanation:
Notice how each congruent square is divided into four equal parts. Since each square has four parts and there are four squares, there are of these parts in total. Because each of these parts make up
of each square, all parts are equal. Therefore, you can count the total number of shaded parts to get your percentage.
The top left and bottom right both have one part shaded each for a total of parts.
The top right and bottom left have and
parts shaded, respectively, for a total of
parts.
Therefore, in total, there are parts shaded out of
total parts.
Thus, the percentage of the total area that is shaded is:
.
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<h3>Two answers: 5, 7</h3>
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Explanation:
A drawing may be helpful to see what's going on. Check out the diagram below. This is one way of drawing out the two triangles. The locations of the points don't really matter, and neither does the the orientation of how you rotate things. What does matter is we have the right points connected to form the segments mentioned.
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For now, focus on triangle TIP only. In order to have this be isosceles, we must make TP = 5 or TP = 7.
If TP = 5, then it's the same length as TI.
If TP = 7, then it's the same length as PI.
In either case, we have exactly two sides the same length (the other side different) which is what it means for a triangle to be isosceles.
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Let's consider triangle TOP. For it to be isosceles, we must have two sides the same length. We already locked in TP to be either 5 or 7 in the previous section above. So there's no way that TP could be 11 units long to match up with PO = 11.
If TP = 5, then OT must also be 5 units long so that triangle TOP is isosceles.
If TP = 7, then OT = 7 for similar reasoning.
Either way, TP only has two choices on what it could be.
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In short, we basically just write the first two values given to us to get the two triangles to be isosceles. We can't use TP = 11 as it would make triangle TIP to be scalene (all sides are different lengths).