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Difference between revisions of "1964 AHSME Problems/Problem 35"

(Created page with "==Problem== The sides of a triangle are of lengths <math>13</math>, <math>14</math>, and <math>15</math>. The altitudes of the triangle meet at point <math>H</math>. if <math>AD...")
 
(→‎Solution 4 (Similar Triangles))
 
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==Problem==
 
==Problem==
  
−
The sides of a triangle are of lengths <math>13</math>, <math>14</math>, and <math>15</math>. The altitudes of the triangle meet at point <math>H</math>. if <math>AD</math> is teh altitude to the side of length <math>14</math>, the ratio <math>HD:HA</math> is:
+
The sides of a triangle are of lengths <math>13</math>, <math>14</math>, and <math>15</math>. The altitudes of the triangle meet at point <math>H</math>. if <math>AD</math> is the altitude to the side of length <math>14</math>, the ratio <math>HD:HA</math> is:
  
 
<math>\textbf{(A) }3:11\qquad\textbf{(B) }5:11\qquad\textbf{(C) }1:2\qquad\textbf{(D) }2:3\qquad \textbf{(E) }25:33</math>
 
<math>\textbf{(A) }3:11\qquad\textbf{(B) }5:11\qquad\textbf{(C) }1:2\qquad\textbf{(D) }2:3\qquad \textbf{(E) }25:33</math>
 +
 +
==Solution 1==
 +
Using Law of Cosines
 +
and the fact that the ratio equals cos(a)/[cos(b)cos(c)]
 +
B 5:11
 +
 +
==Solution 2 (coordinates)==
 +
 +
<asy>
 +
draw((0,0)--(15,0)--(6.6,11.2)--(0,0));
 +
draw((0,0)--(9.6,7.2));
 +
draw((6.6,0)--(6.6,11.2));
 +
draw((15,0)--(3267/845,5544/845));
 +
label("$B$",(15,0),SE);
 +
label("$C$",(6.6,11.2),N);
 +
label("$E$",(6.6,0),S);
 +
label("$15$",(7.5,-0.75),S);
 +
label("$14$",(11,5.75),ENE);
 +
label("$13$",(3,6),WNW);
 +
label("$A$",(0,0),SW);
 +
label("$D$",(9.6,7.2),NE);
 +
label("$H$",(6.6,3.5),E);
 +
</asy>
 +
The reason why we have <math>HD</math> shorter than <math>HA</math> is that all the ratios' left hand side (<math>HD</math>) is less than the ratios' right hand side (<math>HA</math>).
 +
 +
We label point <math>A</math> as the origin and point <math>B</math>, logically, as <math>(15,0)</math>. By Heron's Formula, the area of this triangle is <math>84.</math> Thus the height perpendicular to <math>AB</math> has a length of <math>11.2,</math> and by the Pythagorean Theorem, <math>AE</math> and <math>EB</math> have lengths <math>6.6</math> and <math>8.4,</math> respectively. These lengths tell us that <math>C</math> is at <math>(6.6,11.2)</math>.
 +
 +
The slope of <math>BC</math> is <math>\dfrac{0-11.2}{15-6.6}=-\dfrac{4}{3},</math> and the slope of <math>AD</math> is <math>\dfrac{3}{4}</math> by taking the negative reciprocal of <math>-\dfrac{4}{3}.</math> Therefore, the equation of line <math>AD</math> can best be represented by <math>y=\dfrac{3}{4}x.</math>
 +
 +
We next find the intersection of <math>CE</math> and <math>AD</math>. We automatically know the <math>x</math>-value; it is just <math>6.6</math> because <math>CE</math> is a straight line hitting <math>(6.6,0).</math> Therefore, the <math>y</math>-value is at <math>\dfrac{3}{4}\times 6.6=4.95.</math> Therefore, the intersection between <math>CE</math> and <math>AD</math> is at <math>(6.6,4.95)</math>.
 +
 +
We also need to find the intersection between <math>BC</math> and <math>AD</math>. To do that, we know that the line of <math>AD</math> is represented as <math>y=\dfrac{3}{4}x,</math> and the slope of line <math>BC</math> is <math>-\dfrac{4}{3}.</math> We just need to find line <math>BC</math>'s y-intercept. So far, we have <math>y=-\dfrac{4}{3}x+b,</math> where <math>b</math> is a real y-intercept. We know that <math>B</math> is located at <math>(15,0),</math> so we plug that into the equation and yield <math>b=20.</math> Therefore, the intersection between the two lines is
 +
<cmath>
 +
\dfrac{3}{4}x=-\dfrac{4}{3}x+20,
 +
9x=-16x+240,
 +
25x=240,
 +
x=9.6,
 +
y=\dfrac{3}{4}\times 9.6,
 +
y=7.2.
 +
</cmath>
 +
After that, we use the distance formula: <math>HA</math> has a length of <cmath>\sqrt{(6.6-0)^2+(4.95-0)^2}=\sqrt{\dfrac{1089}{25}+\dfrac{9801}{400}}=\sqrt{\dfrac{1089*16+9801}{400}}=\sqrt{\dfrac{27225}{400}}=\sqrt{\dfrac{1089}{16}}=\dfrac{33}{4}=8.25,</cmath> and <math>HD</math> has a length of <cmath>\sqrt{(9.6-6.6)^2+(7.2-4.95)^2}=\sqrt{3^2+(\dfrac{36}{5}-\dfrac{99}{20})^2}=\sqrt{9+\dfrac{81}{16}}=\sqrt{\dfrac{225}{16}}=3.75.</cmath>
 +
Thus, we have that <math>\dfrac{3.75}{8.25}=\dfrac{\frac{15}{4}}{\frac{33}{4}}=\dfrac{15}{33}=\dfrac{5}{11}=\boxed{\bold{B}}.</math>-OreoChocolate
 +
 +
==Solution 3 (Ceva's Theorem)==
 +
 +
<asy>
 +
size(150);
 +
real a = 14, b = 15, c = 13;
 +
pair C = (0, 0), B = (a, 0);
 +
 +
// calculate cos(α) and sin(α)
 +
real cos_alpha = (a^2 + c^2 - b^2) / (2 * a * c);
 +
real sin_alpha = sqrt(1 - cos_alpha^2);
 +
 +
// calculate coordinates of A
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pair A = (c * cos_alpha, c * sin_alpha);
 +
 +
// calculate altitudes
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pair D = foot(A, B, C);
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pair E = foot(B, A, C);
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pair F = foot(C, A, B);
 +
 +
// calculate orthocenter
 +
pair H = extension(A, D, B, E);
 +
 +
// draw triangle and altitudes
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draw(A--B--C--cycle);
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draw(A--D, dashed);
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draw(B--E, dashed);
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draw(C--F, dashed);
 +
 +
// draw right angle markers
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draw(rightanglemark(A, D, B, 20));
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draw(rightanglemark(B, E, A, 20));
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draw(rightanglemark(C, F, A, 20));
 +
 +
// label points
 +
dot("$A$", A, N);
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dot("$B$", B, SW);
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dot("$C$", C, SE);
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dot("$D$", D, S);
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dot("$E$", E, NW);
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dot("$F$", F, NE);
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dot("$H$", H, SE);
 +
 +
// label side lengths
 +
label("$13$", (A+C)/2, NW);
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label("$15$", (A+B)/2, NE);
 +
label("$14$", (B+C)/2, S);
 +
</asy>
 +
 +
A consequence of Ceva's Theorem that is sometimes attributed to van Aubel states that:
 +
 +
<cmath>\dfrac{AH}{HD} = \dfrac{AE}{EC} + \dfrac{AF}{FB}</cmath>
 +
 +
We must then find <math>AE</math> and <math>AF</math>.  To find <math>AF</math> note that <math>\triangle AFC</math> and <math>\triangle BFC</math> are both right triangles sharing a common height, <math>FC</math>.  Thus
 +
 +
<math>AF^2+FC^2=AC^2, \text{and } BF^2+FC^2=BC^2 \implies</math>
 +
<math>AF^2+FC^2=13^2, \text{and } (15-AF)^2+FC^2=14^2</math>
 +
 +
Subtracting the two equations to eliminate the common height term (<math>FC^2</math>):
 +
 +
<math>(15-AF)^2-AF^2=27 \implies AF=\dfrac{33}{5}</math>
 +
 +
A similar computation using <math>\triangle AEB</math> and <math>\triangle CEB</math> gives us:
 +
 +
<math>AE^2+EB^2=AB^2, \text{and } CE^2+EB^2=BC^2 \implies</math>
 +
<math>AE^2+EB^2=15^2, \text{and } (13-AE)^2+EB^2=14^2</math>
 +
 +
<math>AE^2-(13-AE)^2=29 \implies AE=\dfrac{99}{13}</math>
 +
 +
Returning to our original van Aubel equation yields:
 +
 +
<math>\dfrac{AH}{HD} = \dfrac{AE}{EC} + \dfrac{AF}{FB} = \dfrac{\dfrac{99}{13}}{13-\dfrac{99}{13}} + \dfrac{\dfrac{33}{5}}{15-\dfrac{33}{5}}=\dfrac{99}{70}+\dfrac{55}{70}=\dfrac{11}{5}</math>
 +
 +
Therefore <math>HD:HA = \boxed{\textbf{(B) }5:11}</math>
 +
 +
~ proloto
 +
 +
==Solution 4 (Similar Triangles)==
 +
Because this triangle is a nice triangle, we can easily find that <math>AD = 12</math>, <math>CD = 5</math>, and <math>DB = 9</math>.
 +
From this, we know that:
 +
<math>\triangle BDA</math> is a 3:4:5 triangle, and it is similar to <math>\triangle BEC</math> which is similar to <math>\triangle CDH</math>.
 +
Therefore <math>HD = CD \cdot 3/4 = 15/4</math>, and <math>AH = 12-15/4 = 33/4</math>,
 +
<math>HD/AH = \dfrac{15/4}{33/4} = \dfrac{5}{11}=\boxed{\bold{B}}</math>. -ELIZABETHWZYING
 +
 +
==See Also==
 +
{{AHSME 40p box|year=1964|num-b=34|num-a=36}}
 +
 +
[[Category:Introductory Geometry Problems]]
 +
 +
{{MAA Notice}}

Latest revision as of 11:42, 24 November 2024

Problem

The sides of a triangle are of lengths $13$, $14$, and $15$. The altitudes of the triangle meet at point $H$. if $AD$ is the altitude to the side of length $14$, the ratio $HD:HA$ is:

$\textbf{(A) }3:11\qquad\textbf{(B) }5:11\qquad\textbf{(C) }1:2\qquad\textbf{(D) }2:3\qquad \textbf{(E) }25:33$

Solution 1

Using Law of Cosines and the fact that the ratio equals cos(a)/[cos(b)cos(c)] B 5:11

Solution 2 (coordinates)

[asy] draw((0,0)--(15,0)--(6.6,11.2)--(0,0)); draw((0,0)--(9.6,7.2)); draw((6.6,0)--(6.6,11.2)); draw((15,0)--(3267/845,5544/845)); label("$B$",(15,0),SE); label("$C$",(6.6,11.2),N); label("$E$",(6.6,0),S); label("$15$",(7.5,-0.75),S); label("$14$",(11,5.75),ENE); label("$13$",(3,6),WNW); label("$A$",(0,0),SW); label("$D$",(9.6,7.2),NE); label("$H$",(6.6,3.5),E); [/asy] The reason why we have $HD$ shorter than $HA$ is that all the ratios' left hand side ($HD$) is less than the ratios' right hand side ($HA$).

We label point $A$ as the origin and point $B$, logically, as $(15,0)$. By Heron's Formula, the area of this triangle is $84.$ Thus the height perpendicular to $AB$ has a length of $11.2,$ and by the Pythagorean Theorem, $AE$ and $EB$ have lengths $6.6$ and $8.4,$ respectively. These lengths tell us that $C$ is at $(6.6,11.2)$.

The slope of $BC$ is $\dfrac{0-11.2}{15-6.6}=-\dfrac{4}{3},$ and the slope of $AD$ is $\dfrac{3}{4}$ by taking the negative reciprocal of $-\dfrac{4}{3}.$ Therefore, the equation of line $AD$ can best be represented by $y=\dfrac{3}{4}x.$

We next find the intersection of $CE$ and $AD$. We automatically know the $x$-value; it is just $6.6$ because $CE$ is a straight line hitting $(6.6,0).$ Therefore, the $y$-value is at $\dfrac{3}{4}\times 6.6=4.95.$ Therefore, the intersection between $CE$ and $AD$ is at $(6.6,4.95)$.

We also need to find the intersection between $BC$ and $AD$. To do that, we know that the line of $AD$ is represented as $y=\dfrac{3}{4}x,$ and the slope of line $BC$ is $-\dfrac{4}{3}.$ We just need to find line $BC$'s y-intercept. So far, we have $y=-\dfrac{4}{3}x+b,$ where $b$ is a real y-intercept. We know that $B$ is located at $(15,0),$ so we plug that into the equation and yield $b=20.$ Therefore, the intersection between the two lines is \[\dfrac{3}{4}x=-\dfrac{4}{3}x+20, 9x=-16x+240, 25x=240, x=9.6, y=\dfrac{3}{4}\times 9.6, y=7.2.\] After that, we use the distance formula: $HA$ has a length of \[\sqrt{(6.6-0)^2+(4.95-0)^2}=\sqrt{\dfrac{1089}{25}+\dfrac{9801}{400}}=\sqrt{\dfrac{1089*16+9801}{400}}=\sqrt{\dfrac{27225}{400}}=\sqrt{\dfrac{1089}{16}}=\dfrac{33}{4}=8.25,\] and $HD$ has a length of \[\sqrt{(9.6-6.6)^2+(7.2-4.95)^2}=\sqrt{3^2+(\dfrac{36}{5}-\dfrac{99}{20})^2}=\sqrt{9+\dfrac{81}{16}}=\sqrt{\dfrac{225}{16}}=3.75.\] Thus, we have that $\dfrac{3.75}{8.25}=\dfrac{\frac{15}{4}}{\frac{33}{4}}=\dfrac{15}{33}=\dfrac{5}{11}=\boxed{\bold{B}}.$-OreoChocolate

Solution 3 (Ceva's Theorem)

[asy] size(150); real a = 14, b = 15, c = 13; pair C = (0, 0), B = (a, 0); // calculate cos(α) and sin(α) real cos_alpha = (a^2 + c^2 - b^2) / (2 * a * c); real sin_alpha = sqrt(1 - cos_alpha^2); // calculate coordinates of A pair A = (c * cos_alpha, c * sin_alpha); // calculate altitudes pair D = foot(A, B, C); pair E = foot(B, A, C); pair F = foot(C, A, B); // calculate orthocenter pair H = extension(A, D, B, E); // draw triangle and altitudes draw(A--B--C--cycle); draw(A--D, dashed); draw(B--E, dashed); draw(C--F, dashed); // draw right angle markers draw(rightanglemark(A, D, B, 20)); draw(rightanglemark(B, E, A, 20)); draw(rightanglemark(C, F, A, 20)); // label points dot("$A$", A, N); dot("$B$", B, SW); dot("$C$", C, SE); dot("$D$", D, S); dot("$E$", E, NW); dot("$F$", F, NE); dot("$H$", H, SE); // label side lengths label("$13$", (A+C)/2, NW); label("$15$", (A+B)/2, NE); label("$14$", (B+C)/2, S); [/asy]

A consequence of Ceva's Theorem that is sometimes attributed to van Aubel states that:

\[\dfrac{AH}{HD} = \dfrac{AE}{EC} + \dfrac{AF}{FB}\]

We must then find $AE$ and $AF$. To find $AF$ note that $\triangle AFC$ and $\triangle BFC$ are both right triangles sharing a common height, $FC$. Thus

$AF^2+FC^2=AC^2, \text{and } BF^2+FC^2=BC^2 \implies$ $AF^2+FC^2=13^2, \text{and } (15-AF)^2+FC^2=14^2$

Subtracting the two equations to eliminate the common height term ($FC^2$):

$(15-AF)^2-AF^2=27 \implies AF=\dfrac{33}{5}$

A similar computation using $\triangle AEB$ and $\triangle CEB$ gives us:

$AE^2+EB^2=AB^2, \text{and } CE^2+EB^2=BC^2 \implies$ $AE^2+EB^2=15^2, \text{and } (13-AE)^2+EB^2=14^2$

$AE^2-(13-AE)^2=29 \implies AE=\dfrac{99}{13}$

Returning to our original van Aubel equation yields:

$\dfrac{AH}{HD} = \dfrac{AE}{EC} + \dfrac{AF}{FB} = \dfrac{\dfrac{99}{13}}{13-\dfrac{99}{13}} + \dfrac{\dfrac{33}{5}}{15-\dfrac{33}{5}}=\dfrac{99}{70}+\dfrac{55}{70}=\dfrac{11}{5}$

Therefore $HD:HA = \boxed{\textbf{(B) }5:11}$

~ proloto

Solution 4 (Similar Triangles)

Because this triangle is a nice triangle, we can easily find that $AD = 12$, $CD = 5$, and $DB = 9$. From this, we know that: $\triangle BDA$ is a 3:4:5 triangle, and it is similar to $\triangle BEC$ which is similar to $\triangle CDH$. Therefore $HD = CD \cdot 3/4 = 15/4$, and $AH = 12-15/4 = 33/4$, $HD/AH = \dfrac{15/4}{33/4} = \dfrac{5}{11}=\boxed{\bold{B}}$. -ELIZABETHWZYING

See Also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 34 		Followed by
Problem 36
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

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