Difference between revisions of "2017 AMC 10B Problems/Problem 19"
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==Problem== | ==Problem== | ||
− | Let <math>ABC</math> be an equilateral triangle. Extend side <math>\overline{AB}</math> beyond <math>B</math> to a point <math>B'</math> so that <math>BB'= | + | Let <math>ABC</math> be an equilateral triangle. Extend side <math>\overline{AB}</math> beyond <math>B</math> to a point <math>B'</math> so that <math>BB'=3 \cdot AB</math>. Similarly, extend side <math>\overline{BC}</math> beyond <math>C</math> to a point <math>C'</math> so that <math>CC'=3 \cdot BC</math>, and extend side <math>\overline{CA}</math> beyond <math>A</math> to a point <math>A'</math> so that <math>AA'=3 \cdot CA</math>. What is the ratio of the area of <math>\triangle A'B'C'</math> to the area of <math>\triangle ABC</math>? |
<math>\textbf{(A)}\ 9:1\qquad\textbf{(B)}\ 16:1\qquad\textbf{(C)}\ 25:1\qquad\textbf{(D)}\ 36:1\qquad\textbf{(E)}\ 37:1</math> | <math>\textbf{(A)}\ 9:1\qquad\textbf{(B)}\ 16:1\qquad\textbf{(C)}\ 25:1\qquad\textbf{(D)}\ 36:1\qquad\textbf{(E)}\ 37:1</math> | ||
− | |||
− | == | + | == Diagram == |
− | + | <asy> | |
+ | size(12cm); | ||
+ | real a = 1; | ||
+ | pair A = (0, 0), B = rotate(60)*A + (a, 0), C = rotate(120)*B + (a, 0); | ||
− | + | // calculate extended points | |
− | + | pair A_prime = A + 3*(A - C); | |
+ | pair B_prime = B + 3*(B - A); | ||
+ | pair C_prime = C + 3*(C - B); | ||
− | ===Solution 3 === | + | // draw original triangle and extended triangle |
+ | draw(A--B--C--cycle); | ||
+ | draw(A--A_prime); | ||
+ | draw(B--B_prime); | ||
+ | draw(C--C_prime); | ||
+ | draw(A_prime--B_prime--C_prime--cycle); | ||
+ | |||
+ | // label points | ||
+ | dot("$A$", A, SE); | ||
+ | dot("$B$", B, NE); | ||
+ | dot("$C$", C, W); | ||
+ | dot("$A'$", A_prime, SW); | ||
+ | dot("$B'$", B_prime, NE); | ||
+ | dot("$C'$", C_prime, W); | ||
+ | |||
+ | // label sides | ||
+ | label("$1$", (A+B)/2, S); | ||
+ | label("$1$", (B+C)/2, E); | ||
+ | label("$1$", (C+A)/2, W); | ||
+ | |||
+ | label("$3$", (A+A_prime)/2, E); | ||
+ | label("$3$", (B+B_prime)/2, N); | ||
+ | label("$3$", (C+C_prime)/2, W); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | ==Solution 1 (Uses Trig) == | ||
+ | Note that by symmetry, <math>\triangle A'B'C'</math> is also equilateral. Therefore, we only need to find one of the sides of <math>A'B'C'</math> to determine the area ratio. WLOG, let <math>AB = BC = CA = 1</math>. Therefore, <math>BB' = 3</math> and <math>BC' = 4</math>. Also, <math>\angle B'AC' = 120^{\circ}</math>, so by the Law of Cosines, <math>B'C' = \sqrt{4^2 + 3^2 - 2\cdot(4)(3)\cos(120^\circ)} = \sqrt{16+9 - 24(-1/2)} = \sqrt{37}</math>. Therefore, the answer is <math>(\sqrt{37})^2 : 1^2 = \boxed{\textbf{(E) } 37:1}</math> | ||
+ | |||
+ | ==Solution 2 == | ||
+ | As mentioned in the first solution, <math>\triangle A'B'C'</math> is equilateral. WLOG, let <math>AB=2</math>. Let <math>D</math> be on the line passing through <math>AB</math> such that <math>A'D</math> is perpendicular to <math>AB</math>. Note that <math>\triangle A'DA</math> is a <math>30-60-90</math> with right angle at <math>D</math>. Since <math>AA'=6</math>, <math>AD=3</math> and <math>A'D=3\sqrt{3}</math>. So we know that <math>DB'=11</math>. Note that <math>\triangle A'DB'</math> is a right triangle with right angle at <math>D</math>. So by the Pythagorean theorem, we find <math>A'B'= \sqrt{(3\sqrt{3})^2 + 11^2} = 2\sqrt{37}.</math> Therefore, the answer is <math>(2\sqrt{37})^2 : 2^2 = \boxed{\textbf{(E) } 37:1}</math>. | ||
+ | |||
+ | ==Solution 3 == | ||
Let <math>AB=BC=CA=x</math>. We start by noting that we can just write <math>AB'</math> as just <math>AB+BB'=4AB</math>. | Let <math>AB=BC=CA=x</math>. We start by noting that we can just write <math>AB'</math> as just <math>AB+BB'=4AB</math>. | ||
Similarly <math>BC'=4BC</math>, and <math>CA'=4CA</math>. We can evaluate the area of triangle <math>ABC</math> by simply using Heron's formula, | Similarly <math>BC'=4BC</math>, and <math>CA'=4CA</math>. We can evaluate the area of triangle <math>ABC</math> by simply using Heron's formula, | ||
Line 21: | Line 57: | ||
Similarly from above we know <math>AB'</math> and <math>A'A</math> to be <math>4x</math>, and <math>3x</math> respectively. Thus the area of <math>\Delta{AA'B'}</math> is <math>\frac{1}{2}\cdot 4x\cdot 3x \cdot \sin{120}=3x^2\cdot\sqrt{3}</math>. Likewise we can find <math>BB'C', \text{ and } CC'A'</math> to also be <math>3x^2\cdot\sqrt{3}</math>. | Similarly from above we know <math>AB'</math> and <math>A'A</math> to be <math>4x</math>, and <math>3x</math> respectively. Thus the area of <math>\Delta{AA'B'}</math> is <math>\frac{1}{2}\cdot 4x\cdot 3x \cdot \sin{120}=3x^2\cdot\sqrt{3}</math>. Likewise we can find <math>BB'C', \text{ and } CC'A'</math> to also be <math>3x^2\cdot\sqrt{3}</math>. | ||
<math>[A'B'C']=[AA'B']+[BB'C']+[CC'A']+[ABC]=3\cdot3x^2\cdot\sqrt{3}+\frac{x^2\sqrt{3}}{4}=\sqrt{3}\cdot\Bigg(9x^2+\frac{x^2}{4}\Bigg)</math>. | <math>[A'B'C']=[AA'B']+[BB'C']+[CC'A']+[ABC]=3\cdot3x^2\cdot\sqrt{3}+\frac{x^2\sqrt{3}}{4}=\sqrt{3}\cdot\Bigg(9x^2+\frac{x^2}{4}\Bigg)</math>. | ||
− | Therefore the ratio of <math>[A'B'C']</math> to <math>[ABC]</math> is <math>\frac{\sqrt{3}\cdot\Bigg(9x^2+\frac{x^2}{4}\Bigg)}{\frac{x^2\sqrt{3}}{4}}=\boxed{\textbf{(E) } 37 : 1}</math> | + | Therefore the ratio of <math>[A'B'C']</math> to <math>[ABC]</math> is <math>\frac{\sqrt{3}\cdot\Bigg(9x^2+\frac{x^2}{4}\Bigg)}{\frac{x^2\sqrt{3}}{4}}=\boxed{\textbf{(E) } 37:1}</math> |
− | + | ==Solution 4 (Elimination) == | |
− | Looking at the answer choices, we see that all but <math>{\textbf{(E)}}</math> has a perfect square in the ratio. With some intuition, we can guess that the sidelength of the new triangle formed is not an integer, thus we pick <math>\boxed{\textbf{(E) } 37 : 1}</math>. | + | Looking at the answer choices, we see that all but <math>{\textbf{(E)}}</math> has a perfect square in the ratio. With some intuition, we can guess that the sidelength of the new triangle formed is not an integer, thus we pick <math>\boxed{\textbf{(E) } 37:1}</math>. |
Solution by sp1729 | Solution by sp1729 | ||
− | + | ==Solution 5 (Barycentric Coordinates) == | |
− | + | We use barycentric coordinates wrt <math>\triangle ABC</math>, to which we can easily obtain that <math>A'=(4,0,-3)</math>, <math>B'=(-3,4,0)</math>, and <math>C'=(0,-3,4)</math>. Now, since the coordinates are homogenized (<math>-3+4=1</math>), we can directly apply the area formula to obtain that | |
− | <math>A=( | + | <cmath>[A'B'C']=[ABC]\cdot\left| \begin{array}{ccc} 4 & 0 & -3 \\ -3 & 4 & 0 \\ 0 & -3 & 4 \end{array} \right| = (64-27)[ABC]=37[ABC],</cmath> |
− | + | so the answer is <math>\boxed{\textbf{(E) } 37:1}</math> | |
− | Now since the coordinates are homogenized (<math>-3+4=1</math>), we can | + | |
− | So the | + | ==Solution 6 (Area Comparison) == |
+ | First, comparing bases yields that <math>[BA'B']=3[AA'B]=9[ABC]\implies [AA'B']=12</math>. By congruent triangles, | ||
+ | <cmath>[AA'B']=[BB'C']=[CC'A']\implies [A'B'C']=(12+12+12+1)[ABC],</cmath> | ||
+ | so <math>[A'B'C']:[ABC]=\boxed{\textbf{(E) } 37:1}</math> | ||
+ | |||
+ | == Solution 7 (Quick Proportionality) == | ||
+ | Scale down the figure so that the area formulas for the <math>120^\circ</math> and equilateral triangles become proportional with proportionality constant equivalent to the product of the corresponding sides. By the proportionality, it becomes clear that the answer is <math>3\cdot4\cdot3+1\cdot1=37, \boxed{\text{E}}</math>. | ||
+ | |||
+ | ~ solution by mathchampion1 | ||
+ | |||
+ | ==Solution 8 (Sin area formula) == | ||
+ | Drawing the diagram, we see that the large triangle, <math>A'B'C'</math>, is composed of three congruent triangles with the triangle <math>ABC</math> at the center. Let each of the sides of triangle <math>ABC</math> be <math>x</math>. Therefore, using the equilateral triangle area formula, the <math>[ABC] = \frac{x^2\sqrt{3}}{4}</math>. We also know now that the sides of the triangles are <math>3x</math> and <math>3x + x</math>, or <math>4x</math>. We also know that since <math>BB'</math> are collinear, as are the others, angle <math>C'BB'</math> is <math>180 - 60</math>, which is <math>120</math> degrees. Because that angle is an included angle, we get the area of all three congruent triangle's are <math>\frac{12x^2\sin120}{2} \cdot 3</math>. Simplifying that yields <math>\frac{36x^2\sqrt{3}}{4}</math>. Adding that to the <math>[ABC]</math> yields <math>\frac{37x^2\sqrt{3}}{4}</math>. From this, we can compare the ratios by canceling everything out except for the <math>37</math>, so the answer is <math>\boxed{\textbf{(E) }37:1}</math> | ||
+ | |||
+ | ~Solution by EricShi1685 | ||
+ | |||
+ | |||
+ | ==Solution 9 (Same as Solution 8 but faster)== | ||
+ | WLOG, let the side length of the smaller triangle be 1. The area of the big portion (A'B'A) is then <math>\frac{1}{2}\cdot3\cdot4\cdot\sin\left(120\right)=\frac{1}{2}\cdot3\cdot4\cdot\sin\left(60\right)=\frac{1}{2}\cdot3\cdot4\cdot\frac{\sqrt{3}}{2}=3\sqrt{3}</math>. Now simply multiply by three and add <math>\frac{\sqrt{3}}{4}</math> (the area of the small triangle) we get <math>\frac{37\sqrt{3}}{4}</math> and so the ratio is <math>37:1</math>. <math>\boxed{\textbf{(E) }37:1}</math> | ||
+ | |||
+ | ==Solution 10 (Area Ratios)== | ||
+ | Connect <math>BA', CB',</math> and <math>AC'</math>. Let <math>[\triangle ABC]=k</math>. <math>\frac{[\triangle AC'C]}{[\triangle ABC]}=\frac{CC'}{BC}=3</math>. So the area of <math>\triangle ACC'</math> is equal to <math>3k</math>. We can do the same thing for <math>\triangle AA'C'</math>. <math>\frac{[\triangle A'AC']}{[\triangle ACC']}=\frac{A'A}{AC}=3</math>. Thus, the area of <math>\triangle AA'C'</math> is equal to <math>9k</math>. We will now find the area of <math>\triangle A'B'C'</math> in terms of <math>k</math>. <math>[\triangle A'B'C']=[\triangle ABC]+[\triangle A'CC']+[\triangle B'AA']+[\triangle C'BB']=[\triangle ABC]+3[\triangle A'CC']</math>. The area of <math>\triangle A'CC'</math> is equal to the sum of the areas of <math>\triangle ACC'</math> and <math>\triangle AA'C'</math>, which is <math>12k</math>. So the area of <math>\triangle A'B'C'</math> is equal to <math>37k</math> and the area of <math>\triangle ABC</math> is equal to <math>k</math> so the ratio of the area of <math>\triangle A'B'C'</math> to the area of <math>\triangle ABC</math> is equal to <math>\boxed{\textbf{(E) } 37:1}</math> | ||
+ | -Heavytoothpaste | ||
+ | |||
+ | |||
+ | ==Solution 11 (Quick Guess If You Have No Time)== | ||
+ | <math>(A)</math>, <math>(B)</math>, <math>(C)</math>, and <math>(D)</math> are all perfect squares, which makes them seem unlikely, so we can guess that the answer probably is <math>\boxed{(E)}</math> as it is the only one with a side length that is implied to not be an integer. | ||
+ | |||
+ | == Solution 12 (Mass Points and Routh) == | ||
+ | |||
+ | This looks like an easy application of Routh's Theorem, except we are only given information about the ratios of the cevians, not the side lengths of <math>\triangle A'B'C'</math>. | ||
+ | |||
+ | Let's figure those out. Extend <math>A'A</math> and <math>B'B</math> to meet <math>B'C'</math> and <math>A'C'</math> at <math>D</math> and <math>E</math>, respectively (Note we only need to look at 2 of 3 cevians to figure everything out). Call the unknown lengths <math>CD=AE=x</math> so our new diagram showing the cevian ratios is as follows. | ||
+ | |||
+ | <asy> | ||
+ | size(250); | ||
+ | real a = 1; | ||
+ | pair A = (0, 0), B = rotate(60)*A + (a, 0), C = rotate(120)*B + (a, 0); | ||
+ | pair shiftVector = (8, 0); // define a shift vector to the right | ||
+ | |||
+ | // calculate extended points | ||
+ | pair A_prime = A + 3*(A - C); | ||
+ | pair B_prime = B + 3*(B - A); | ||
+ | pair C_prime = C + 3*(C - B); | ||
+ | pair D = extension(A, A_prime, B_prime, C_prime); | ||
+ | pair EE = extension(B, B_prime, A_prime, C_prime); | ||
+ | |||
+ | // draw original triangle and extended triangle | ||
+ | draw(A_prime--D); | ||
+ | draw(B_prime--EE); | ||
+ | draw(A_prime--B_prime--C_prime--cycle); | ||
+ | |||
+ | // label points | ||
+ | dot("$A$", A, SE); | ||
+ | dot("$D$", D, NE); | ||
+ | dot("$E$", EE, W); | ||
+ | dot("$A'$", A_prime, SW); | ||
+ | dot("$B'$", B_prime, NE); | ||
+ | dot("$C'$", C_prime, W); | ||
+ | |||
+ | // label sides | ||
+ | label(scale(0.8)*"$4$", (A+B_prime)/2, S); | ||
+ | label(scale(0.8)*"$x+1$", (A+D)/2, E); | ||
+ | label(scale(0.8)*"$3$", (A+A_prime)/2, E); | ||
+ | label(scale(0.8)*"$x$", (A+EE)/2, S); | ||
+ | |||
+ | // draw duplicated triangle and labels shifted to the right | ||
+ | draw(shift(shiftVector)*A_prime--shift(shiftVector)*D); | ||
+ | draw(shift(shiftVector)*B_prime--shift(shiftVector)*EE); | ||
+ | draw(shift(shiftVector)*A_prime--shift(shiftVector)*B_prime--shift(shiftVector)*C_prime--cycle); | ||
+ | |||
+ | // label cevian ratios | ||
+ | label(scale(0.8)*"$4$", shift(2*shiftVector)*(A+B_prime)/2, S); | ||
+ | label(scale(0.8)*"$x+1$", shift(2*shiftVector)*(A+D)/2, E); | ||
+ | label(scale(0.8)*"$3$", shift(2*shiftVector)*(A+A_prime)/2, E); | ||
+ | label(scale(0.8)*"$x$", shift(2*shiftVector)*(A+EE)/2, S); | ||
+ | |||
+ | // mass point labels | ||
+ | pair mA = shift(shiftVector)*A + scale(0.5)*SE; | ||
+ | label(scale(0.4)*"$x+4$", mA, UnFill); | ||
+ | draw(Circle(mA, .4), linewidth(1)); | ||
+ | pair mB_prime = shift(shiftVector)*B_prime + scale(0.5)*E; | ||
+ | label(scale(0.4)*"$x$", mB_prime, UnFill); | ||
+ | draw(Circle(mB_prime, .4), linewidth(1)); | ||
+ | pair mE = shift(shiftVector)*EE + scale(0.5)*W; | ||
+ | label(scale(0.4)*"$4$", mE, UnFill); | ||
+ | draw(Circle(mE, .4), linewidth(1)); | ||
+ | pair mA_prime = shift(shiftVector)*A_prime + scale(0.5)*SW; | ||
+ | label(scale(0.4)*"$x+1$", mA_prime, UnFill); | ||
+ | draw(Circle(mA_prime, .4), linewidth(1)); | ||
+ | pair mD = shift(shiftVector)*D + scale(0.5)*NE; | ||
+ | label(scale(0.4)*"$3$", mD, UnFill); | ||
+ | draw(Circle(mD, .4), linewidth(1)); | ||
+ | pair mC_prime = shift(shiftVector)*C_prime + scale(0.5)*NW; | ||
+ | label(scale(0.4)*"$3-x$", mC_prime, UnFill); | ||
+ | draw(Circle(mC_prime, .4), linewidth(1)); | ||
+ | |||
+ | |||
+ | // draw double right arrow between original and duplicated diagram | ||
+ | draw((4.5,0)--(5.3,0),EndArrow(5)); | ||
+ | </asy> | ||
+ | |||
+ | with the balanced mass points on the right. Now by the symmetry in the original diagram, <math>\dfrac{A'E}{EC'}=\dfrac{C'D}{DB'}</math> so: | ||
+ | |||
+ | <cmath>\dfrac{3-x}{x+1}=\dfrac{x}{3-x} \implies x=\dfrac{9}{7} \implies \dfrac{A'E}{EC'}=\dfrac{3}{4}</cmath> | ||
+ | |||
+ | Finally we can apply Routh's Theorem: | ||
+ | |||
+ | <cmath> | ||
+ | \dfrac{[\triangle ABC]}{[\triangle A'B'C']} = \frac{(x y z-1)^2}{(x y+y+1) (x z+x+1) (y z+z+1)} = \frac{(\frac{37}{64})^2}{(\frac{37}{16})^3} = \frac{1}{37} | ||
+ | </cmath> | ||
+ | |||
+ | Hence <math>[\triangle A'B'C']:[\triangle ABC] = \boxed{\textbf{(E) } 37:1}</math> | ||
+ | |||
+ | ~ proloto | ||
− | Solution by | + | == Video Solution by OmegaLearn (Law of Cosines)== |
+ | https://youtu.be/4_x1sgcQCp4?t=5373 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | == Video Solution by OmegaLearn (Meta-Solving Technique) == | ||
+ | https://youtu.be/GmUWIXXf_uk?t=710 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2017|ab=B|num-b=18|num-a=20}} | {{AMC10 box|year=2017|ab=B|num-b=18|num-a=20}} | ||
{{AMC12 box|year=2017|ab=B|num-b=14|num-a=16}} | {{AMC12 box|year=2017|ab=B|num-b=14|num-a=16}} | ||
+ | |||
+ | [[Category:Introductory Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 23:42, 17 July 2023
Contents
- 1 Problem
- 2 Diagram
- 3 Solution 1 (Uses Trig)
- 4 Solution 2
- 5 Solution 3
- 6 Solution 4 (Elimination)
- 7 Solution 5 (Barycentric Coordinates)
- 8 Solution 6 (Area Comparison)
- 9 Solution 7 (Quick Proportionality)
- 10 Solution 8 (Sin area formula)
- 11 Solution 9 (Same as Solution 8 but faster)
- 12 Solution 10 (Area Ratios)
- 13 Solution 11 (Quick Guess If You Have No Time)
- 14 Solution 12 (Mass Points and Routh)
- 15 Video Solution by OmegaLearn (Law of Cosines)
- 16 Video Solution by OmegaLearn (Meta-Solving Technique)
- 17 See Also
Problem
Let be an equilateral triangle. Extend side
beyond
to a point
so that
. Similarly, extend side
beyond
to a point
so that
, and extend side
beyond
to a point
so that
. What is the ratio of the area of
to the area of
?
Diagram
Solution 1 (Uses Trig)
Note that by symmetry, is also equilateral. Therefore, we only need to find one of the sides of
to determine the area ratio. WLOG, let
. Therefore,
and
. Also,
, so by the Law of Cosines,
. Therefore, the answer is
Solution 2
As mentioned in the first solution, is equilateral. WLOG, let
. Let
be on the line passing through
such that
is perpendicular to
. Note that
is a
with right angle at
. Since
,
and
. So we know that
. Note that
is a right triangle with right angle at
. So by the Pythagorean theorem, we find
Therefore, the answer is
.
Solution 3
Let . We start by noting that we can just write
as just
.
Similarly
, and
. We can evaluate the area of triangle
by simply using Heron's formula,
.
Next in order to evaluate
we need to evaluate the area of the larger triangles
.
In this solution we shall just compute
of these as the others are trivially equivalent.
In order to compute the area of
we can use the formula
.
Since
is equilateral and
,
,
are collinear, we already know
Similarly from above we know
and
to be
, and
respectively. Thus the area of
is
. Likewise we can find
to also be
.
.
Therefore the ratio of
to
is
Solution 4 (Elimination)
Looking at the answer choices, we see that all but has a perfect square in the ratio. With some intuition, we can guess that the sidelength of the new triangle formed is not an integer, thus we pick
.
Solution by sp1729
Solution 5 (Barycentric Coordinates)
We use barycentric coordinates wrt , to which we can easily obtain that
,
, and
. Now, since the coordinates are homogenized (
), we can directly apply the area formula to obtain that
so the answer is
Solution 6 (Area Comparison)
First, comparing bases yields that . By congruent triangles,
so
Solution 7 (Quick Proportionality)
Scale down the figure so that the area formulas for the and equilateral triangles become proportional with proportionality constant equivalent to the product of the corresponding sides. By the proportionality, it becomes clear that the answer is
.
~ solution by mathchampion1
Solution 8 (Sin area formula)
Drawing the diagram, we see that the large triangle, , is composed of three congruent triangles with the triangle
at the center. Let each of the sides of triangle
be
. Therefore, using the equilateral triangle area formula, the
. We also know now that the sides of the triangles are
and
, or
. We also know that since
are collinear, as are the others, angle
is
, which is
degrees. Because that angle is an included angle, we get the area of all three congruent triangle's are
. Simplifying that yields
. Adding that to the
yields
. From this, we can compare the ratios by canceling everything out except for the
, so the answer is
~Solution by EricShi1685
Solution 9 (Same as Solution 8 but faster)
WLOG, let the side length of the smaller triangle be 1. The area of the big portion (A'B'A) is then . Now simply multiply by three and add
(the area of the small triangle) we get
and so the ratio is
.
Solution 10 (Area Ratios)
Connect and
. Let
.
. So the area of
is equal to
. We can do the same thing for
.
. Thus, the area of
is equal to
. We will now find the area of
in terms of
.
. The area of
is equal to the sum of the areas of
and
, which is
. So the area of
is equal to
and the area of
is equal to
so the ratio of the area of
to the area of
is equal to
-Heavytoothpaste
Solution 11 (Quick Guess If You Have No Time)
,
,
, and
are all perfect squares, which makes them seem unlikely, so we can guess that the answer probably is
as it is the only one with a side length that is implied to not be an integer.
Solution 12 (Mass Points and Routh)
This looks like an easy application of Routh's Theorem, except we are only given information about the ratios of the cevians, not the side lengths of .
Let's figure those out. Extend and
to meet
and
at
and
, respectively (Note we only need to look at 2 of 3 cevians to figure everything out). Call the unknown lengths
so our new diagram showing the cevian ratios is as follows.
with the balanced mass points on the right. Now by the symmetry in the original diagram, so:
Finally we can apply Routh's Theorem:
Hence
~ proloto
Video Solution by OmegaLearn (Law of Cosines)
https://youtu.be/4_x1sgcQCp4?t=5373
~ pi_is_3.14
Video Solution by OmegaLearn (Meta-Solving Technique)
https://youtu.be/GmUWIXXf_uk?t=710
~ pi_is_3.14
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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All AMC 12 Problems and Solutions |
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