Logo for LIVE by Po-Shen Loh

LIVE

English

2017 AMC 12A Problem 19

Attempt Problem 19 of the 2017 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

19.

A square with side length xx is inscribed in a right triangle with sides of length 3,3, 4,4, and 55 so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length yy is inscribed in another right triangle with sides of length 3,3, 4,4, and 55 so that one side of the square lies on the hypotenuse of the triangle. What is xy?\dfrac{x}{y}?

1213\dfrac{12}{13}

3537\dfrac{35}{37}

11

3735\dfrac{37}{35}

1312\dfrac{13}{12}

Answer: D
Concepts:similarityright triangle
Difficulty rating: 2040
Small Hint:

A square with its corner at the right angle has side x=3⋅43+4x=\dfrac{3\cdot4}{3+4}

Big Hint:

A square with a side on a base of length bb and height hh to that base has side bhb+h;\dfrac{bh}{b+h}; use the hypotenuse as the base

Solution:

For the first square, the two smaller triangles it cuts off are similar to the whole triangle, giving x3−x=4−xx,\dfrac{x}{3-x}=\dfrac{4-x}{x}, so x=127.x=\dfrac{12}{7}. (Equivalently, a square in the right angle has side 3⋅43+4.\dfrac{3\cdot4}{3+4}.)

For the second square, take the hypotenuse of length 55 as base; the altitude to it is h=3⋅45=125.h=\dfrac{3\cdot4}{5}=\dfrac{12}{5}. A square with a side on a base bb and height hh has side bhb+h,\dfrac{bh}{b+h}, so y=5⋅1255+125=12375=6037. y=\dfrac{5\cdot\tfrac{12}{5}}{5+\tfrac{12}{5}}=\dfrac{12}{\tfrac{37}{5}}=\dfrac{60}{37}.

Therefore xy=127⋅3760=3735.\dfrac{x}{y}=\dfrac{12}{7}\cdot\dfrac{37}{60}=\dfrac{37}{35}.

Thus, the correct answer is D.

Problem 18#18
Full Exam
Problem 20#20

Problem 19 in Other Years

1950 AMC 12 · 1951 AMC 12 · 1952 AMC 12 · 1953 AMC 12 · 1954 AMC 12 · 1955 AMC 12 · 1956 AMC 12 · 1957 AMC 12 · 1958 AMC 12 · 1959 AMC 12 · 1960 AMC 12 · 1961 AMC 12 · 1962 AMC 12 · 1963 AMC 12 · 1964 AMC 12 · 1965 AMC 12 · 1966 AMC 12 · 1967 AMC 12 · 1968 AMC 12 · 1969 AMC 12 · 1970 AMC 12 · 1971 AMC 12 · 1972 AMC 12 · 1973 AMC 12 · 1974 AMC 12 · 1975 AMC 12 · 1976 AMC 12 · 1977 AMC 12 · 1978 AMC 12 · 1979 AMC 12 · 1980 AMC 12 · 1981 AMC 12 · 1982 AMC 12 · 1983 AMC 12 · 1984 AMC 12 · 1985 AMC 12 · 1986 AMC 12 · 1987 AMC 12 · 1988 AMC 12 · 1989 AMC 12 · 1990 AMC 12 · 1991 AMC 12 · 1992 AMC 12 · 1993 AMC 12 · 1994 AMC 12 · 1995 AMC 12 · 1996 AMC 12 · 1997 AMC 12 · 1998 AMC 12 · 1999 AMC 12 · 2000 AMC 12 · 2001 AMC 12 · 2002 AMC 12A · 2002 AMC 12B · 2003 AMC 12A · 2003 AMC 12B · 2004 AMC 12A · 2004 AMC 12B · 2005 AMC 12A · 2005 AMC 12B · 2006 AMC 12A · 2006 AMC 12B · 2007 AMC 12A · 2007 AMC 12B · 2008 AMC 12A · 2008 AMC 12B · 2009 AMC 12A · 2009 AMC 12B · 2010 AMC 12A · 2010 AMC 12B · 2011 AMC 12A · 2011 AMC 12B · 2012 AMC 12A · 2012 AMC 12B · 2013 AMC 12A · 2013 AMC 12B · 2014 AMC 12A · 2014 AMC 12B · 2015 AMC 12A · 2015 AMC 12B · 2016 AMC 12A · 2016 AMC 12B · 2017 AMC 12B · 2018 AMC 12A · 2018 AMC 12B · 2019 AMC 12A · 2019 AMC 12B · 2020 AMC 12A · 2020 AMC 12B · 2021 AMC 12A Spring · 2021 AMC 12B Spring · 2021 AMC 12A Fall · 2021 AMC 12B Fall · 2022 AMC 12A · 2022 AMC 12B · 2023 AMC 12A · 2023 AMC 12B · 2024 AMC 12A · 2024 AMC 12B · 2025 AMC 12A · 2025 AMC 12B