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### Course: Geometry (all content)>Unit 4

Lesson 4: Perpendicular bisectors

# 2003 AIME II problem 7

Find the area of

rhombus ABCD given that the radii of the circles

circumscribed about triangles ABD and ACD

.
Created by Sal Khan.

## Want to join the conversation?

• Aime is for high school right?
• Mostly, yes. AIME is short for American Invitational Mathematics Examination. The "invitational" part is because you have to qualify to take it, either by scoring highly (usually around 98th percentile or higher - there is a set score, but it varies year to year) on the AMC 10/12 (which are open to anyone who is not above that grade level - that is, you can take it in middle school, although it is intended for people in high school) or by getting the highest score in your state on the AMCs. If you score highly on the AIME, which has fifteen questions of increasing difficulty, you take the USAMO, United States of America Mathematical Olympiad, which consists of 6 problems given in two 4.5 - hour sets of three. Of the top twelve high scorers from this, six go on to represent the USA in the International Mathematical Olympiad, or IMO. AIME problems and solutions can be found at http://www.artofproblemsolving.com/Wiki/index.php/Aime_problems
• does AIME stand for somthing
• It stands for American Invitational Mathematics Examination.
• What is a rhombus?
• According to my on-line dictionary, a "RHOMBUS" is "an equilateral parallelogram, including the square as a special case." In other words, a rhombus is a 4-sided figure, with all 4 sides of the same length, where the angles do not have to be (but can be) right angles. Hope this helps. Good Luck.
• Ths video refers to several other videos about triangles and competition geometry. It seems to have been set up from another video, which wasn't directly before the video. I would be great if there were video links to videos to whihc the lesson refers. It was very confusing and startling to have something with such an increased level of complexity with very little preparation.
• at Sal says that the diameter of the circles circumscribed around the triangles abd and acd are 12.5 and 25 but the text says radii
• Yeah, he did the wording wrong, but he still did indeed solve the problem correctly.
• Misran Rehman and I have the same question. The "10 = a" should be "10 = a^2", which would make the end steps a little more complicated.
(1 vote)
• No. He says around that 50a=5a^2. 5a^2 is 5*a*a. So dividing by 5a on both sides, 50a/5a = 10. 5*a*a/5*a=a so 10=a, not a^2
• what is a circumscribed circle?
• Wiki said: " the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon"
In another word, all corners of a polygon are on its circumscribed circle.
Each Triangle, rectangle and regular polygon have its one and only one circumscribed circle. Other polygons may or may not have circumscribed circles.
I hope that helps.
• at , he said 50=5a^2 , then he divided by 5 and got 10=a. what about that squared symbol?
• The equation was 50a=5a^2. He then took each side divided by 5a, which came out to be 10=a, or a=10.
• What does AIME stand for?
(1 vote)
• AIME stands for the American Invitational Mathematics Examination.
• At Sal divides by 5a. we know diving by a is legal since a is not equal to zero. but how can we know it doesn't make us lose solutions or something similar? how do we know it's a totally legit action?
(1 vote)
• Precisely because a is not equal to zero. That's the only algebraic solution you'd lose here by making that move.