Difference between revisions of "1991 AHSME Problems/Problem 19"

Latest revision as of 21:41, 23 November 2020

Problem

$[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP("C",(0,0),SW);MP("A",(0,3),NW);MP("B",(4,0),S);MP("E",(7,0),SE);MP("D",(7,10),NE); [/asy]$

Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$ . Triangle $ABD$ has a right angle at $A$ and $AD=12$ . Points $C$ and $D$ are on opposite sides of $\overline{AB}$ . The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$ . If $\frac{DE}{DB}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, then $m+n=$

$\text{(A) } 25\quad \text{(B) } 128\quad \text{(C) } 153\quad \text{(D) } 243\quad \text{(E) } 256$

Solution 1

Solution by e_power_pi_times_i

Let $F$ be the point such that $DF$ and $CF$ are parallel to $CE$ and $DE$ , respectively, and let $DE = x$ and $BE^2 = 169-x^2$ . Then, $[FDEC] = x(4+\sqrt{169-x^2}) = [ABC] + [BED] + [ABD] + [AFD] = 6 + \dfrac{x\sqrt{169-x^2}}{2} + 30 + \dfrac{(x-3)(4+\sqrt{169-x^2})}{2}$ . So, $4x+x\sqrt{169-x^2} = 60 + x\sqrt{169-x^2} - 3\sqrt{169-x^2}$ . Simplifying $3\sqrt{169-x^2} = 60 - 4x$ , and $1521 - 9x^2 = 16x^2 - 480x + 3600$ . Therefore $25x^2 - 480x + 2079 = 0$ , and $x = \dfrac{48\pm15}{5}$ . Checking, $x = \dfrac{63}{5}$ is the answer, so $\dfrac{DE}{DB} = \dfrac{\dfrac{63}{5}}{13} = \dfrac{63}{65}$ . The answer is $\boxed{\textbf{(B) } 128}$ .

Solution 2

Solution by Arjun Vikram

Extend lines $AD$ and $CE$ to meet at a new point $F$ . Now, we see that $FAC\sim FDE \sim ACB$ . Using this relationship, we can see that $AF=\frac{15}4$ , (so $FD=\frac{63}4$ ), and the ratio of similarity between $FDE$ and $FAC$ is $\frac{63}{15}$ . This ratio gives us that $\frac{63}5$ . By the Pythagorean Theorem, $DB=13$ . Thus, $\frac{DE}{DB}=\frac{63}{65}$ , and the answer is $63+65=\boxed{\textbf{(B) } 128}$ .

Solution 3 (Trig)

We have $\angle ABC = \arcsin\left(\frac{3}{5}\right)$ and $\angle DBA=\arcsin\left(\frac{12}{13}\right).$ Now we are trying to find $\sin(\angle DBE)=\sin\left(180^{\circ}-\angle DBC=\sin(180^{\circ}-\arcsin\left(\frac{3}{5}\right)\right)-\arcsin\left(\frac{12}{13}\right)=\sin(\arcsin\left(\frac{3}{5}\right)+\arcsin\left(\frac{12}{13}\right)).$ Now we use the $\sin$ angle sum identity, which states $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b).$ Using this identity yields $\sin\left(\arcsin\left(\frac{3}{5}\right)\right)\cos\left(\arcsin\left(\frac{12}{13}\right)\right)+\cos\left(\arcsin\left(\frac{3}{5}\right)\right)+\sin\left(\arcsin\left(\frac{12}{13}\right)\right).$

@@ Line 17: / Line 17: @@
 \text{(E) } 256</math>
-== Solution ==
+== Solution 1 ==
 Solution by e_power_pi_times_i
-Let <math>F</math> be the point such that <math>DF</math> and <math>CF</math> are parallel to <math>CE</math> and <math>DE</math>, respectively, and let <math>DE = x</math> and <math>BE^2 = 169-x^2</math>. Then, <math>[FDEC] = x(4+\sqrt{169-x^2}) = [ABC] + [BED] + [ABD] + [AFD] = 6 + \dfrac{x\sqrt{169-x^2}}{2} + 30 + \dfrac{(x-3)(4+\sqrt{169-x^2})}{2}</math>. So, <math>4x+x\sqrt{169-x^2}) = 60 + x\sqrt{169-x^2} - 3\sqrt{169-x^2}</math>. Simplifying <math>3\sqrt{169-x^2} = 60 - 4x</math>, and <math>1521 - 9x^2 = 16x^2 - 480x + 3600</math>. Therefore <math>25x^2 - 480x + 2079 = 0</math>, and <math>x = \dfrac{48\pm15}{5}</math>. Checking, <math>x = \dfrac{63}{5}</math> is the answer, so <math>\dfrac{DE}{DB} = \dfrac{\dfrac{63}{5}}{13} = \dfrac{63}{65}</math>. The answer is <math>\boxed{\textbf{(B) } 128}</math>.
+Let <math>F</math> be the point such that <math>DF</math> and <math>CF</math> are parallel to <math>CE</math> and <math>DE</math>, respectively, and let <math>DE = x</math> and <math>BE^2 = 169-x^2</math>. Then, <math>[FDEC] = x(4+\sqrt{169-x^2}) = [ABC] + [BED] + [ABD] + [AFD] = 6 + \dfrac{x\sqrt{169-x^2}}{2} + 30 + \dfrac{(x-3)(4+\sqrt{169-x^2})}{2}</math>. So, <math>4x+x\sqrt{169-x^2} = 60 + x\sqrt{169-x^2} - 3\sqrt{169-x^2}</math>. Simplifying <math>3\sqrt{169-x^2} = 60 - 4x</math>, and <math>1521 - 9x^2 = 16x^2 - 480x + 3600</math>. Therefore <math>25x^2 - 480x + 2079 = 0</math>, and <math>x = \dfrac{48\pm15}{5}</math>. Checking, <math>x = \dfrac{63}{5}</math> is the answer, so <math>\dfrac{DE}{DB} = \dfrac{\dfrac{63}{5}}{13} = \dfrac{63}{65}</math>. The answer is <math>\boxed{\textbf{(B) } 128}</math>.
+== Solution 2 ==
+Solution by Arjun Vikram
+Extend lines <math>AD</math> and <math>CE</math> to meet at a new point <math>F</math>. Now, we see that <math>FAC\sim FDE \sim ACB</math>. Using this relationship, we can see that <math>AF=\frac{15}4</math>, (so <math>FD=\frac{63}4</math>), and the ratio of similarity between <math>FDE</math> and <math>FAC</math> is <math>\frac{63}{15}</math>. This ratio gives us that <math>\frac{63}5</math>. By the Pythagorean Theorem, <math>DB=13</math>. Thus, <math>\frac{DE}{DB}=\frac{63}{65}</math>, and the answer is <math>63+65=\boxed{\textbf{(B) } 128}</math>.
+== Solution 3 (Trig)==
+We have <math>\angle ABC = \arcsin\left(\frac{3}{5}\right)</math> and <math>\angle DBA=\arcsin\left(\frac{12}{13}\right).</math> Now we are trying to find <math>\sin(\angle DBE)=\sin\left(180^{\circ}-\angle DBC=\sin(180^{\circ}-\arcsin\left(\frac{3}{5}\right)\right)-\arcsin\left(\frac{12}{13}\right)=\sin(\arcsin\left(\frac{3}{5}\right)+\arcsin\left(\frac{12}{13}\right)).</math> Now we use the <math>\sin</math> angle sum identity, which states <math>\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b).</math> Using this identity yields <math>\sin\left(\arcsin\left(\frac{3}{5}\right)\right)\cos\left(\arcsin\left(\frac{12}{13}\right)\right)+\cos\left(\arcsin\left(\frac{3}{5}\right)\right)+\sin\left(\arcsin\left(\frac{12}{13}\right)\right).</math>
 == See also ==

1991 AHSME (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 • 26 • 27 • 28 • 29 • 30
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