Direction & Angles | PRMO-2019 | Problem 4

Try this beautiful Geometry problem from PRMO, 2019 based on Direction & Angles.

Direction & Angles | PRMO | Problem-4

An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160° turn to the right and walks 4 more feet. It then makes another 160° turn to the right and walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is the distance in feet it would have walked?

Key Concepts

Geometry

Co-ordinate geometry

Trigonometry

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Answer:$36$

PRMO-2018, Problem 13

Pre College Mathematics

Try with Hints

According to the problem we draw the figure and try to solve using co-ordinate method...

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

Let $A_0(0,0)$

Therefore $A_1(4 cos 0,4sin 0)$

$A_2(4cos 0+4 cos 160, 4 sin 0+4 sin 160)$

$\Rightarrow A_n =(0,0)$

and $4(cos 0+cos 160 +...+sin 160(n-1)=0$

$\Rightarrow n =9$

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

Therefore distance covered =$4 \times 9$=$36$

Medians | Geometry | PRMO-2018 | Problem 13

Try this beautiful Geometry problem from PRMO, 2018 based on Medians.

Medians | PRMO | Problem-13

In a triangle ABC, right-angled at A, the altitude through A and the internal bisector of $\angle A $ have lengths $3$ and $4$, respectively. Find the length of the median through $A$.

$20$
$24$
$13$

Key Concepts

Geometry

Medians

Triangle

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Answer:$24$

PRMO-2018, Problem 13

Pre College Mathematics

Try with Hints

Now in the Right angle Triangle ABC ,$\angle A=90$.$AD=3$ is Altitude and $AE=4$ is the internal Bisector and $AF$ is the median.Now we have to find out the length of $AF$

Now $\angle CAE = 45^{\circ }= \angle BAE$.

Let$ BC = a$, $CA = b$, $AB = c$
so $\frac{bc}{2}=\frac{3a}{2}$ $\Rightarrow bc=3a$

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

Therefore,

$\frac{2bc}{b+c} cos\frac{A}{2}=4$

$\Rightarrow \frac{6a}{b+c} .\frac{1}{\sqrt 2}=4$

$\Rightarrow 2\sqrt 2 (b+c)=3a$

$\Rightarrow 8(b^2 + c^2 + 2bc) = 9a^2$

$\Rightarrow 8(a^2 + 6a) = 9a^2$

$\Rightarrow 48a=a^2$

$\Rightarrow a=48$

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

Now $AF$ is the median , $AF=\frac{a}{2}=24$

Direction & Angles | PRMO-2019 | Problem 4

Direction & Angles | PRMO | Problem-4

Key Concepts

Check the Answer

Try with Hints

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Medians | Geometry | PRMO-2018 | Problem 13

Medians | PRMO | Problem-13

Key Concepts

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Try with Hints

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Direction & Angles | PRMO | Problem-4

Key Concepts

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Try with Hints

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Medians | PRMO | Problem-13

Key Concepts

Check the Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube