What is the NO-SHORTCUT approach for learning great Mathematics?

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Try this beautiful problem from Geometry: Area of a triangle

A circle of radius 1 is tangent to a circle of radius 2 . The sides of $\triangle A B C$ are tangent to the circles as shown, and the sides $\overline{A B}$ and $\overline{A C}$ are congruent. What is the area of $\triangle A B C ?$

,

i

- $15 \sqrt{2}$
- $\frac{35}{2} $
- $\frac{64}{3}$
- $16 \sqrt{2}$
- \(24\)

Geometry

Circle

Triangle

But try the problem first...

Answer: $16 \sqrt{2}$

Source

Suggested Reading

AMC-10A (2006) Problem 21

Pre College Mathematics

First hint

Given that there are two circle of radius 1 is tangent to a circle of radius 2.we have to find out the area of the \(\triangle ABC\).Now draw a perpendicular line \(AF\) on \(BC\).Clearly it will pass through two centers \(O_1\) and \(O_2\). and $\overline{A B}$ and $\overline{A C}$ are congruent i.e \(\triangle ABC\) is an Isosceles triangle. Therefore \(BF=FC\)

So if we can find out \(AF\) and \(BC\) then we can find out the area of the \(\triangle ABC\).can you find out \(AF\) and \(BC\)?

Can you now finish the problem ..........

Second Hint

Now clearly $\triangle A D O_{1} \sim \triangle A E O_{2} \sim \triangle A F C$ ( as \(O_1D\) and \(O_2E\) are perpendicular on \(AC\) , R-H-S law )

From Similarity we can say that , $\frac{A O_{1}}{A O_{2}}=\frac{D O_{1}}{E O_{2}} \Rightarrow \frac{A O_{1}}{A O_{1}+3}=\frac{1}{2} \Longrightarrow A O_{1}=3$

By the Pythagorean Theorem we have that $A D=\sqrt{3^{2}-1^{2}}=\sqrt{8}$

Again from $\triangle A D O_{1} \sim \triangle A F C$

$\frac{A D}{A F}=\frac{D O_{1}}{C F} \Longrightarrow \frac{2 \sqrt{2}}{8}=\frac{1}{C F} \Rightarrow C F=2 \sqrt{2}$

can you finish the problem........

Final Step

The area of the triangle is $\frac{1}{2} \cdot A F \cdot B C=\frac{1}{2} \cdot A F \cdot(2 \cdot C F)=A F \cdot C F=8(2 \sqrt{2})$=\(16\sqrt2\)

- https://www.cheenta.com/surface-area-of-cube-amc-10a-2007-problem-21/
- https://www.youtube.com/watch?v=gGT15ls_brU

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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