Difference between revisions of "1990 AIME Problems/Problem 7"

Latest revision as of 11:34, 22 November 2023

Problem

A triangle has vertices $P_{}^{}=(-8,5)$ , $Q_{}^{}=(-15,-19)$ , and $R_{}^{}=(1,-7)$ . The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0_{}^{}$ . Find $a+c_{}^{}$ .

$[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17); MP("P",P,N,f);MP("Q",Q,W,f);MP("R",R,E,f); D(P--Q--R--cycle);D(P--T,EndArrow(2mm)); D((-17,0)--(4,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm)); [/asy]$

Solution

Use the distance formula to determine the lengths of each of the sides of the triangle. We find that it has lengths of side $15,\ 20,\ 25$ , indicating that it is a $3-4-5$ right triangle. At this point, we just need to find another point that lies on the bisector of $\angle P$ .

Solution 1

$[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17),U=IP(P--T,Q--R); MP("P",P,N,f);MP("Q",Q,W,f);MP("R",R,E,f);MP("P'",U,SE,f); D(P--Q--R--cycle);D(U);D(P--U); D((-17,0)--(4,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm)); [/asy]$

Use the angle bisector theorem to find that the angle bisector of $\angle P$ divides $QR$ into segments of length $\frac{25}{x} = \frac{15}{20 -x} \Longrightarrow x = \frac{25}{2},\ \frac{15}{2}$ . It follows that $\frac{QP'}{RP'} = \frac{5}{3}$ , and so $P' = \left(\frac{5x_R + 3x_Q}{8},\frac{5y_R + 3y_Q}{8}\right) = (-5,-23/2)$ .

The desired answer is the equation of the line $PP'$ . $PP'$ has slope $\frac{-11}{2}$ , from which we find the equation to be $11x + 2y + 78 = 0$ . Therefore, $a+c = \boxed{089}$ .

Solution 2

$[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17); MP("P",P,N,f);MP("Q",Q,W,f);MP("R",R,NE,f);MP("S",S,E,f); D(P--Q--R--cycle);D(R--S--Q,dashed);D(T);D(P--T); D((-17,0)--(4,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm)); [/asy]$

Extend $PR$ to a point $S$ such that $PS = 25$ . This forms an isosceles triangle $PQS$ . The coordinates of $S$ , using the slope of $PR$ (which is $-4/3$ ), can be determined to be $(7,-15)$ . Since the angle bisector of $\angle P$ must touch the midpoint of $QS \Rightarrow (-4,-17)$ , we have found our two points. We reach the same answer of $11x + 2y + 78 = 0$ .

Solution 3

$[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17),U=IP(P--T,Q--R); MP("P",P,N,f);MP("Q",Q,W,f);MP("R",R,E,f);MP("P'",U,SE,f); D(P--Q--R--cycle);D(U);D(P--U); D((-17,0)--(4,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm)); D(Q--(U.x,Q.y)--U,dashed);D(rightanglemark(Q,(U.x,Q.y),U,20),dashed); [/asy]$

By the angle bisector theorem as in solution 1, we find that $QP' = 25/2$ . If we draw the right triangle formed by $Q, P',$ and the point directly to the right of $Q$ and below $P'$ , we get another $3-4-5 \triangle$ (since the slope of $QR$ is $3/4$ ). Using this, we find that the horizontal projection of $QP'$ is $10$ and the vertical projection of $QP'$ is $15/2$ .

Thus, the angle bisector touches $QR$ at the point $\left(-15 + 10, -19 + \frac{15}{2}\right) = \left(-5,-\frac{23}{2}\right)$ , from where we continue with the first solution.

Solution 4

This solution uses terminology from the other solutions. The incenter is a much easier point to find on the line $PP'$ . Note that the inradius of $\triangle PQR$ is $5$ . If you do not understand this, substitute values into the $[\triangle ABC] = rs$ equation. If lines are drawn from the incenter perpendicular to $PR$ and $QR$ , then a square with side length $5$ will be created. Call the point opposite $R$ in this square $R'$ . Since $R$ has coordinates $(1, -7)$ , and the sides of the squares are on a $3-4-5$ ratio, the coordinates of $R'$ are $(-6, -6)$ . This is because the x-coordinate is moving to the left $4+3=7$ units and the y-coordinate is moving up $-3+4=1$ units. The line through $(-8,5)$ and $(-6,-6)$ is $11x+2y+78=0$ .

Solution 5 (Trigonometry)

$[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(0,0),Q=(-7,-24),R=(9,-12),S=(15,-20),T=(4,-22); MP("Q",Q,W,f);MP("R",R,E,f); D(P--Q--R--cycle);D(P--T,EndArrow(2mm)); D((-8,0)--(13,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm)); [/asy]$

Transform triangle $PQR$ so that $P$ is at the origin. Note that the slopes do not change when we transform the triangle.

We know that the slope of $PQ$ is $\frac{24}{7}$ and the slope of $PR$ is $-\frac{4}{3}$ . Thus, in the complex plane, they are equivalent to $\tan(\alpha)=\frac{24}{7}$ and $\tan(\beta)=-\frac{4}{3}$ , respectively. Here $\alpha$ is the angle formed by the $x$ -axis and $PQ$ , and $\beta$ is the angle formed by the $x$ -axis and $PR$ . The equation of the angle bisector is $\tan\left(\frac{\alpha+\beta}{2}\right)$ .

As the tangents are in very neat Pythagorean triples, we can easily calculate $\cos(\alpha)$ and $\cos(\beta)$ .

Angle $\alpha$ is in the third quadrant, so $\cos(\alpha)$ is negative. Thus $\cos(\alpha)=-\frac{7}{25}$ .

Angle $\beta$ is in the fourth quadrant, so $\cos(\beta)$ is positive. Thus $\cos(\beta)=\frac{3}{5}$ .

By the Half-Angle Identities, $\tan\left(\frac{\alpha}{2}\right)=\pm\sqrt{\frac{1-\cos(\alpha)}{1+\cos(\alpha)}}=\pm\sqrt{\frac{\frac{32}{25}}{\frac{18}{25}}}=\pm\sqrt{\frac{16}{9}}=\pm\frac{4}{3}$ and $\tan\left(\frac{\beta}{2}\right)=\pm\sqrt{\frac{1-\cos(\beta)}{1+\cos(\beta)}}=\pm\sqrt{\frac{\frac{2}{5}}{\frac{8}{5}}}=\pm\sqrt{\frac{1}{4}}=\pm\frac{1}{2}$ .

Since $\frac{\alpha}{2}$ and $\frac{\beta}{2}$ must be in the second quadrant, their tangent values are both negative. Thus $\tan\left(\frac{\alpha}{2}\right)=-\frac{4}{3}$ and $\tan\left(\frac{\beta}{2}\right)=-\frac{1}{2}$ .

By the sum of tangents formula, $\tan\left(\frac{\alpha}{2}+\frac{\beta}{2}\right)=\frac{\tan\left(\frac{\alpha}{2}\right)+\tan\left(\frac{\beta}{2}\right)}{1-\tan\left(\frac{\alpha}{2}\right)\tan\left(\frac{\beta}{2}\right)}=\frac{-\frac{11}{6}}{\frac{1}{3}}=-\frac{11}{2}$ , which is the slope of the angle bisector.

Finally, the equation of the angle bisector is $y-5=-\frac{11}{2}(x+8)$ or $y=-\frac{11}{2}x-39$ . Rearranging, we get $11x+2y+78=0$ , so our sum is $a+c=11+78=\boxed{089}$ . ~eevee9406

@@ Line 52: / Line 52: @@
 Thus, the angle bisector touches <math>QR</math> at the point <math>\left(-15 + 10, -19 + \frac{15}{2}\right) = \left(-5,-\frac{23}{2}\right)</math>, from where we continue with the first solution.
 === Solution 4 ===
+This solution uses terminology from the other solutions. The incenter is a much easier point to find on the line <math>PP'</math>. Note that the inradius of <math>\triangle PQR</math> is <math>5</math>. If you do not understand this, substitute values into the <math>[\triangle ABC] = rs</math> equation. If lines are drawn from the incenter perpendicular to <math>PR</math> and <math>QR</math>, then a square with side length <math>5</math> will be created. Call the point opposite <math>R</math> in this square <math>R'</math>. Since <math>R</math> has coordinates <math>(1, -7)</math>, and the sides of the squares are on a <math>3-4-5</math> ratio, the coordinates of <math>R'</math> are <math>(-6, -6)</math>. This is because the x-coordinate is moving to the left <math>4+3=7</math> units and the y-coordinate is moving up <math>-3+4=1</math> units. The line through <math>(-8,5)</math> and <math>(-6,-6)</math> is <math>11x+2y+78=0</math>.
+===Solution 5 (Trigonometry)===
 <center><asy>
 import graph;
 pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10);
-pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17),U=IP(P--T,Q--R);
+pair P=(0,0),Q=(-7,-24),R=(9,-12),S=(15,-20),T=(4,-22);
-MP("P",P,N,f);MP("Q",Q,W,f);MP("R",R,E,f);MP("P'",U,SE,f);
+MP("Q",Q,W,f);MP("R",R,E,f);
-D(P--Q--R--cycle);D(U);D(P--U);
+D(P--Q--R--cycle);D(P--T,EndArrow(2mm));
-D((-17,0)--(4,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm));
+D((-8,0)--(13,0),Arrows(2mm));D((0,-21)--(0,7),Arrows(2mm));
-label("$X_1$",(-9.5,1),W);
-label("$X_2$",(-3,1),W);
 </asy></center>
-Let <math>X_1</math>, <math>X_2</math> be the x-intercepts of <math>PQ</math>, <math>PR</math>, respectively. Calculating the slopes and plugging in the coordinates, we have the equations of <math>PQ</math> and <math>PR</math> respectively
+Transform triangle <math>PQR</math> so that <math>P</math> is at the origin. Note that the slopes do not change when we transform the triangle.
-<cmath>y=\frac{24}{7}x+\frac{227}{7}</cmath>
-<cmath>y=-\frac{4}{3}x-\frac{17}{3}</cmath>
-Plugging in <math>y=0</math> gets the coordinates of the x-intercepts <math>X_1=(-\frac{227}{24},0)</math>, <math>X_2=(-\frac{17}{4},0)</math>.
+We know that the slope of <math>PQ</math> is <math>\frac{24}{7}</math> and the slope of <math>PR</math> is <math>-\frac{4}{3}</math>. Thus, in the complex plane, they are equivalent to <math>\tan(\alpha)=\frac{24}{7}</math> and <math>\tan(\beta)=-\frac{4}{3}</math>, respectively. Here <math>\alpha</math> is the angle formed by the <math>x</math>-axis and <math>PQ</math>, and <math>\beta</math> is the angle formed by the <math>x</math>-axis and <math>PR</math>. The equation of the angle bisector is <math>\tan\left(\frac{\alpha+\beta}{2}\right)</math>.
+As the tangents are in very neat [[pythagorean triple|Pythagorean triples]], we can easily calculate <math>\cos(\alpha)</math> and <math>\cos(\beta)</math>.
+Angle <math>\alpha</math> is in the third quadrant, so <math>\cos(\alpha)</math> is negative. Thus <math>\cos(\alpha)=-\frac{7}{25}</math>.
+Angle <math>\beta</math> is in the fourth quadrant, so <math>\cos(\beta)</math> is positive. Thus <math>\cos(\beta)=\frac{3}{5}</math>.
+By the [[Trigonometric identities#Half-angle identities|Half-Angle Identities]], <math>\tan\left(\frac{\alpha}{2}\right)=\pm\sqrt{\frac{1-\cos(\alpha)}{1+\cos(\alpha)}}=\pm\sqrt{\frac{\frac{32}{25}}{\frac{18}{25}}}=\pm\sqrt{\frac{16}{9}}=\pm\frac{4}{3}</math> and <math>\tan\left(\frac{\beta}{2}\right)=\pm\sqrt{\frac{1-\cos(\beta)}{1+\cos(\beta)}}=\pm\sqrt{\frac{\frac{2}{5}}{\frac{8}{5}}}=\pm\sqrt{\frac{1}{4}}=\pm\frac{1}{2}</math>.
+Since <math>\frac{\alpha}{2}</math> and <math>\frac{\beta}{2}</math> must be in the second quadrant, their tangent values are both negative. Thus <math>\tan\left(\frac{\alpha}{2}\right)=-\frac{4}{3}</math> and <math>\tan\left(\frac{\beta}{2}\right)=-\frac{1}{2}</math>.
+By the [[Trigonometric identities#Angle addition identities|sum of tangents formula]], <math>\tan\left(\frac{\alpha}{2}+\frac{\beta}{2}\right)=\frac{\tan\left(\frac{\alpha}{2}\right)+\tan\left(\frac{\beta}{2}\right)}{1-\tan\left(\frac{\alpha}{2}\right)\tan\left(\frac{\beta}{2}\right)}=\frac{-\frac{11}{6}}{\frac{1}{3}}=-\frac{11}{2}</math>, which is the slope of the angle bisector.
+Finally, the equation of the angle bisector is <math>y-5=-\frac{11}{2}(x+8)</math> or <math>y=-\frac{11}{2}x-39</math>. Rearranging, we get <math>11x+2y+78=0</math>, so our sum is <math>a+c=11+78=\boxed{089}</math>. ~eevee9406
 == See also ==

1990 AIME (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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