computational with altitudes and midpoints (2009 HOMC Junior Q10)

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0 replies

jlacosta

Apr 2, 2025

0 replies

inequalities - 5/4

pennypc123456789 1

N 20 minutes ago by Mathzeus1024

Given real numbers $x, y$ satisfying $|x| \le 3, |y| \le 3$ . Prove that:
$0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.$

1 reply

pennypc123456789

2 hours ago

Mathzeus1024

20 minutes ago

.problem.

Cobedangiu 3

N an hour ago by Cobedangiu

Find the integer coefficients after expanding Newton's binomial:
$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$

3 replies

Cobedangiu

Yesterday at 6:20 AM

Cobedangiu

an hour ago

Source of a combinatorics problem

isodynamicappolonius2903 0

4 hours ago

Does anyone know exactly the source of this problem?? I just remember that it from a combinatorics book of Russia.

0 replies

isodynamicappolonius2903

4 hours ago

0 replies

KSEA NMSC Mock Contest Group B (Last Problem)

Shiyul 5

N Today at 3:26 AM by Shiyul

Let $a_n$ be a sequence defined by $a_n = a^2 + 1$ . Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$ . Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$ .

5 replies

Shiyul

Yesterday at 3:09 PM

Shiyul

Today at 3:26 AM

No more topics!

computational with altitudes and midpoints (2009 HOMC Junior Q10)

parmenides51 2

N Jul 18, 2019 by amar_04

Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$ . Find the distance between the midpoints of $AD$ and $BC$

2 replies

parmenides51

Jul 18, 2019

amar_04

Jul 18, 2019

computational with altitudes and midpoints (2009 HOMC Junior Q10)

G H J

Reveal topic tags

geometry

altitudes

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parmenides51

30629 posts

parmenides51

#1 h Jul 18, 2019, 2:03 PM • 1 Y

Y by Adventure10

Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$ . Find the distance between the midpoints of $AD$ and $BC$

This post has been edited 1 time. Last edited by parmenides51, Jul 18, 2019, 2:17 PM
Reason: name typos

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Bashy99

698 posts

Bashy99

#2 h Jul 18, 2019, 2:21 PM • 2 Y

Y by Adventure10, Mango247

Solution.

Suppose $E,\,F,\,P$ are the midpoints of $AD,\,BC,\,AC$ respectively. Then $EP=\frac{CD}{2}=\frac{3}{2}$ and $EP||CD.$ Also $PF=\frac{AB}{2}=2$ and $PF||AB.$ Since $CD\perp AB,$ we have $\angle EPF=90^{\circ}.$ Applying Pythagoras' theorem on $\triangle EPF,$ we get $EF=\frac{5}{2}=2.5.$

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amar_04

1915 posts

amar_04

#3 h Jul 18, 2019, 2:23 PM • 3 Y

Y by Path_to_Almighty, Adventure10, Mango247

Another Solution

Extend $BA$ to a point $K$ such that $AK=BD$ . So, $B=AD+DB=AD+AK=DK=4\implies KC=5$ . Now let $M$ be the mid point of $AD$ and $N$ be the midpoint of $BC$ . Then $DM=AM\implies DM+BD=AM+AK\implies BM=KM$ . So, $M$ is also the midpoint of $BK$ too. Hence, $MN=\frac{5}{2}=2.5$ .

This problem is similar to this https://artofproblemsolving.com/community/c6h56249_find_the_distance.
The OP's problem has a homothety with a scale factor of $\frac{1}{2}$ with the problem in the link.

This post has been edited 11 times. Last edited by amar_04, Jul 18, 2019, 2:37 PM

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