Link of the Day
Did you know that you can only see color in the central part of your vision? The rest of your vision is actually black and white. Your brain remembers the colors of items that you have looked at and fills them in. Physicist turned webcomic writer Randall Munroe has a great diagram explaining some of the little-known facts about your vision. You can view the diagram here.
7/11 Triangles
Read the following question and then select the correct answer.
Whenever you are given a diagram, check to see if there is a note that says "figure not drawn to scale." If you don't see a note, that means that the figure is drawn as accurately as possible. In other words, you can look at the diagram and make estimates. On the other hand, if the figure is not drawn to scale, then it is distorted intentionally to trick you. Focus on the facts you know about the figure and not its appearance.
In the figure above, the circle with center 
and the circle with center 
are tangent at point 
. If the circles each have radius 
, and if line 
is tangent to the circle with center 
at point 
, what is the value of 
?
and the circle with center are tangent at point . If the circles each have radius , and if line is tangent to the circle with center at point , what is the value of ?
The diagram above can look a little intimidating. That makes it even more important that you follow the Knowsys method step by step. First, read the problem carefully and identify the bottom line. You are looking for the value of x. Now think about the different ways you could attack the problem. Notice that the figure is drawn to scale (since there is no note that says "figure not drawn to scale"). You could estimate the value of the angle. However, if you look at the answer choices below, they are so close together that an estimation won't really do you much good. Instead, it's best to fill in pieces of the diagram step by step until you can find the value of x. Don't forget to "attack the problem". A positive attitude can make a big difference when you are working a challenging math problem.
You know that the radius of both circles is 10. That means that line AB is 10 and line AC is 20. You also know that line l is tangent to the circle at point B. If you remember your geometry rules for circles, you know that angle ABC is 90 degrees. Now, you need to remember the special triangles. You know that a right triangle with a leg of length x and a hypotenuse of length 2x is a 30-60-90 triangle. That means angle x must be 60 degrees. Don't forget to loop back and verify that your answer matches the bottom line.
(A) 55
(B) 60
(C) 63
(D) 65
(E) It cannot be determined from the information given.
You know that the radius of both circles is 10. That means that line AB is 10 and line AC is 20. You also know that line l is tangent to the circle at point B. If you remember your geometry rules for circles, you know that angle ABC is 90 degrees. Now, you need to remember the special triangles. You know that a right triangle with a leg of length x and a hypotenuse of length 2x is a 30-60-90 triangle. That means angle x must be 60 degrees. Don't forget to loop back and verify that your answer matches the bottom line.
(A) 55
(B) 60
(C) 63
(D) 65
(E) It cannot be determined from the information given.
The correct answer choice is B.
On sat.collegeboard.org 51% of the responses were correct.
For more help with math, visit www.myknowsys.com.
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