BE : ED = GE : EF = 5 : 3 = 3 : FE
EF = 9/5 = BH The answer
Wednesday, May 14, 2025
Similar Triangles: Team question : Beginning level
9. In the figure below, quadrilateral CDEG is a square with CD = 3, and quadrilateral BEFH is a rectangle. If EB = 5, how many units is BH? Express your answer as a mixed number
Triangle BED is a 3-4-5 right triangle and is similar to triangle GEF.
Wednesday, May 7, 2025
Ay reflection notes
Ay
5/7
1) Reviewed the attached problems and solutions and I understand them well
2) Attempted the 2024 AMC 12A in 60 minutes. I attempted #1 to #19 and got them all correct.
I had a headache after that so I couldn't attempt the rest of the questions. Skipped -#20 to #25
Not sure how many I could have actually done.
Monday, February 17, 2025
Harder Mathcounts State/AMC Questions: Intermediate level if you can solve in less than 2 mins.
2012 Mathcounts State Sprint #30: In rectangle ABCD, shown here, point M is the midpoint of side BC, and point N lies on CD such that DN:NC = 1:4. Segment BN intersects AM and AC at points R and S, respectively. If NS:SR:RB = x:y:z, where x, y and z are
positive integers, what is the minimum possible value of x + y + z?
Solution I :
\(\overline {AB}:\overline {NC}=5:4\) [given]
Triangle ASB is similar to triangle CSN (AAA)
\(\overline {NS}:\overline {SB}= 4 : 5\)
Let \(\overline {NS}= 4a, \overline {SB}= 5a.\)
Draw a parallel line to \(\overline {NC}\) from M and mark the interception to \(\overline {BN}\)as T.
\(\overline {MT}: \overline {NC}\) = 1 to 2. [\(\Delta BMT\) and \(\Delta BCN\) are similar triangles ]
\(\overline {NT} = \overline {TB}= \dfrac {4a+5a} {2}=4.5a\)
\(\overline {ST} = 0.5a\)
\(\overline {MT} : \overline {AB}\) = 2 to 5
[Previously we know \(\overline {MT}: \overline {NC}\) = 1 to 2 or 2 to 4 and \(\overline {NC}:\overline {AB}= 4 : 5\) so the ratio of the two lines \(\overline {MT} : \overline {AB}\) is 2 to 5.]
\(\overline {TB} = 4.5 a\) [from previous conclusion]
Using 5 to 2 line ratio [similar triangles \(\Delta ARB\) and \(\Delta MRT\) , you get \(\overline {BR} =\dfrac {5} {7}\times 4.5a =\dfrac {22.5a} {7}\) and \(\overline {RT} =\dfrac {2} {7}\times 4.5a =\dfrac {9a} {7}\)
Thus, x : y : z = 4a : \( \dfrac {1} {2}a + \dfrac {9a} {7}\) : \(\dfrac {22.5a} {7}\) = 56 : 25 : 45
x + y + z = 126
Solution II :
From Mathcounts Mini: Similar Triangles and Proportional Reasoning
Solution III:
Using similar triangles ARB and CRN , you have \(\dfrac {x} {y+z}=\dfrac {5} {9}\).
9x = 5y + 5z ---- equation I
Using similar triangles ASB and CSN and you have \( \dfrac {x+y} {z}=\dfrac {5} {4}\).
4x + 4y = 5z ---- equation II
Plug in (4x + 4y) for 5z on equation I and you have 9x = 5y + (4x + 4y) ; 5x = 9y ; x = \(\dfrac {9} {5}y\)
Plug in x = \(\dfrac {9} {5}y\) to equation II and you have z = \( \dfrac {56} {25}y\)
x : y : z = \(\dfrac {9} {5}y\) : y : \( \dfrac {56} {25}y\) = 45 y : 25y : 56y
45 + 25 + 56 = 126
Solution IV : Yes, there is another way that I've found even faster, saved for my private students. :D
Solution V : from Abhinav, one of my students solving another similar question :
Solution I :
\(\overline {AB}:\overline {NC}=5:4\) [given]
Triangle ASB is similar to triangle CSN (AAA)
\(\overline {NS}:\overline {SB}= 4 : 5\)
Let \(\overline {NS}= 4a, \overline {SB}= 5a.\)
\(\overline {MT}: \overline {NC}\) = 1 to 2. [\(\Delta BMT\) and \(\Delta BCN\) are similar triangles ]
\(\overline {NT} = \overline {TB}= \dfrac {4a+5a} {2}=4.5a\)
\(\overline {ST} = 0.5a\)
\(\overline {MT} : \overline {AB}\) = 2 to 5
[Previously we know \(\overline {MT}: \overline {NC}\) = 1 to 2 or 2 to 4 and \(\overline {NC}:\overline {AB}= 4 : 5\) so the ratio of the two lines \(\overline {MT} : \overline {AB}\) is 2 to 5.]
\(\overline {TB} = 4.5 a\) [from previous conclusion]
Using 5 to 2 line ratio [similar triangles \(\Delta ARB\) and \(\Delta MRT\) , you get \(\overline {BR} =\dfrac {5} {7}\times 4.5a =\dfrac {22.5a} {7}\) and \(\overline {RT} =\dfrac {2} {7}\times 4.5a =\dfrac {9a} {7}\)
Thus, x : y : z = 4a : \( \dfrac {1} {2}a + \dfrac {9a} {7}\) : \(\dfrac {22.5a} {7}\) = 56 : 25 : 45
x + y + z = 126
Solution II :
From Mathcounts Mini: Similar Triangles and Proportional Reasoning
Solution III:
Using similar triangles ARB and CRN , you have \(\dfrac {x} {y+z}=\dfrac {5} {9}\).
9x = 5y + 5z ---- equation I
Using similar triangles ASB and CSN and you have \( \dfrac {x+y} {z}=\dfrac {5} {4}\).
4x + 4y = 5z ---- equation II
Plug in (4x + 4y) for 5z on equation I and you have 9x = 5y + (4x + 4y) ; 5x = 9y ; x = \(\dfrac {9} {5}y\)
Plug in x = \(\dfrac {9} {5}y\) to equation II and you have z = \( \dfrac {56} {25}y\)
x : y : z = \(\dfrac {9} {5}y\) : y : \( \dfrac {56} {25}y\) = 45 y : 25y : 56y
45 + 25 + 56 = 126
Solution IV : Yes, there is another way that I've found even faster, saved for my private students. :D
Solution V : from Abhinav, one of my students solving another similar question :
Two other similar questions from 2016 AMC A, B tests :
2016 AMC 10 A, #19 : Solution from Abhinav
2016 AMC 10 B #19 : Solution from Abhinav
Tuesday, November 12, 2024
A Skill for the 21st Century: Problem Solving by Richard Rusczyk
Does our approach to teaching math fail even the smartest kids ?
Quotes from that article "According to research from the University of California, Los Angeles, as many as 60 percent of all college students who intend to study a STEM (science, technology, engineering, math) subject end up transferring out. In an era when politicians and educators are beside themselves with worry over American students’ lagging math and science scores compared to the whiz kids of Shanghai and Japan, this attrition trend so troubles experts it has spawned an entire field of research on “STEM drop-out,” citing reasons from gender and race to GPAs and peer relationships."
A Skill for the 21st Century: Problem Solving by Richard Rusczyk, founder of "Art of Problem Solving".
Top 10 Skills We Wish Were Taught at School, But Usually Aren't
from Lifehacker
Quotes from that article "According to research from the University of California, Los Angeles, as many as 60 percent of all college students who intend to study a STEM (science, technology, engineering, math) subject end up transferring out. In an era when politicians and educators are beside themselves with worry over American students’ lagging math and science scores compared to the whiz kids of Shanghai and Japan, this attrition trend so troubles experts it has spawned an entire field of research on “STEM drop-out,” citing reasons from gender and race to GPAs and peer relationships."
A Skill for the 21st Century: Problem Solving by Richard Rusczyk, founder of "Art of Problem Solving".
Top 10 Skills We Wish Were Taught at School, But Usually Aren't
from Lifehacker
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