[ad_1]
Problem 1
Thanks to Ismot from Bangladesh for the suggestion!
A right triangle ABC has legs AB = 4, BC = 6, and has a semicircle O with its center on the hypotenuse tangent to the legs AB at D and BC at E. What is the radius of the semicircle?
Problem 2
Thanks Michal from the Czech Republic for suggesting a problem that inspired this puzzle.
What is the value of x in the figure below?
Problem 3
This problem comes from the Bangladesh Mathematical Olympiad (BDMO) 2016.
Two large circles are tangent to each other and the parallel lines. Two small circles are tangent to the large circles and to one of the lines. If the distance between the centers of the small circles is 96, what is the distance between the centers of the large circles?
As usual, watch the video for a solution.
3 Problems About Finding The Length In A Given Figure
Or keep reading.
.
.
“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
.
.
.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
E
.
.
.
.
Answer To Semicircle on Right Triangle Hypotenuse
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
Problem 1
We will prove a more general formula. Let r be the radius of the semicircle. Then we have:
1/r = 1/AB + 1/BC
r = (AB)(BC)/(AB + BC)
The area of the entire right triangle is (1/2)(AB)(BC). Construct radii to the points of tangency.
The area of the entire triangle is also the sum of areas of the triangles AOB and BOC. Each has a height equal to r, and the base is a side of the right triangle. So we have:
area = (1/2)r(AB) + (1/2)r(BC)
Equating the two areas of ABC gives:
(1/2)(AB)(BC) = (1/2)r(AB) + (1/2)r(BC)
(AB)(BC) = r(AB + BC)
r = (AB)(BC)/(AB + BC)
Taking the reciprocal and simplifying gives:
1/r = 1/AB + 1/BC
The original question had legs of 4 and 6, so we have:
r
= 4(6)/(4 + 6)
= 12/5
= 2.4
Problem 2
Extend the slanted lines to construct rectangles and isosceles right triangles. The base consists of 2 hypotenuses of isosceles right triangles with legs of 6 and 4, so their lengths will be 6√2 and 4√2. The other part of the base is a leg of an isosceles triangle with a a leg equal to 10, so it is 10.
The total length is then equal to:
6√2 + 4√2 + 10
= 10√2 + 10
≈ 24.14
Problem 3
Circles that are tangent have their centers and the tangent point collinear. Let the large circles have a radius of a and the small circles b. Construct a right triangle between the center of a large circle, a small circle, and the tangent point of the two large circles. One leg will be a, the other leg will be a – b, and the hypotenuse will be a + b. The distance between the two small circle centers is 2a – 2b, which is given to be 96.
From the diagram and given information we have:
2a – 2b = 96
a2 + (a – b)2 = (a + b)2
From the first equation we have:
a – b = 48
b = a – 48
We can substitute this into the second equation.
a2 + (a – b)2 = (a + b)2
a2 + 482 = (a + a – 48)2
a2 + 482 = (2a – 48)2
a2 + 482 = 4a2 – 192a + 482
0 = 3a2 – 192a
0 = a(3a – 192)
Since a is non-zero, we have:
3a – 192 = 0
a = 64
We want the distance between the two large circles, and that is 2a = 128.
Reference
Problem 1 Math StackExchange
https://math.stackexchange.com/questions/2285180/prove-frac1r-frac1a-frac1b-for-a-semicircle-tangent-within
MY BOOKS
If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.
Book ratings are from January 2023.
(US and worldwide links)
https://mindyourdecisions.com/blog/my-books
Mind Your Decisions is a compilation of 5 books:
(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers By Drawing Lines
The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.3/5 stars on 290 reviews)
40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 54 reviews)
The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.1/5 stars on 33 reviews)
The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 116 reviews)
Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 37 reviews)
Mind Your Puzzles is a collection of the three “Math Puzzles” books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.
Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 112 reviews.
Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.2/5 stars on 33 reviews)
Math Puzzles Volume 3 is the third in the series. (rated 4.2/5 stars on 29 reviews)
KINDLE UNLIMITED
Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.
Currently you can read most of my ebooks through Amazon’s “Kindle Unlimited” program. Included in the subscription you will get access to millions of ebooks. You don’t need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.
US, list of my books (US)
UK, list of my books (UK)
Canada, book results (CA)
Germany, list of my books (DE)
France, list of my books (FR)
India, list of my books (IN)
Australia, book results (AU)
Italy, list of my books (IT)
Spain, list of my books (ES)
Japan, list of my books (JP)
Brazil, book results (BR)
Mexico, book results (MX)
MERCHANDISE
Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.
[ad_2]