Difference between revisions of "2002 AMC 12A Problems/Problem 18"

Latest revision as of 02:13, 1 September 2023

Problem

Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$ ?

$\text{(A) }15 \qquad \text{(B) }18 \qquad \text{(C) }20 \qquad \text{(D) }21 \qquad \text{(E) }24$

Solution 1

First examine the formula $(x-10)^2+y^2=36$ , for the circle $C_1$ . Its center, $D_1$ , is located at (10,0) and it has a radius of $\sqrt{36}$ = 6. The next circle, using the same pattern, has its center, $D_2$ , at (-15,0) and has a radius of $\sqrt{81}$ = 9. So we can construct this diagram: $[asy] unitsize(0.3cm); defaultpen(0.8); path C1=circle((10,0),6); path C2=circle((-15,0),9); draw(C1); draw(C2); draw( (-25,0) -- (17,0) ); dot((10,0)); dot((-15,0)); pair[] p1 = intersectionpoints(C1, circle((5,0),5) ); pair[] p2 = intersectionpoints(C2, circle((-7.5,0),7.5) ); dot(p1[1]); dot(p2[0]); draw((10,0)--p1[1]--p2[0]--(-15,0)); label("$C_1$",(10,0) + 6*dir(-45), SE ); label("$C_2$",(-15,0) + 9*dir(225), SW ); label("$D_1$",(10,0), SE ); label("$D_2$",(-15,0), SW ); label("$Q$", p2[0], NE ); label("$P$", p1[1], SW ); label("$O$", (0,0), SW ); [/asy]$ Line PQ is tangent to both circles, so it forms a right angle with the radii (6 and 9). This, as well as the two vertical angles near O, prove triangles $D_2QO$ and $D_1PO$ similar by AA, with a scale factor of $6:9$ , or $2:3$ . Next, we must subdivide the line $D_2D_1$ in a 2:3 ratio to get the length of the segments $D_2O$ and $D_1O$ . The total length is $10 - (-15)$ , or $25$ , so applying the ratio, $D_2O$ = 15 and $D_1O$ = 10. These are the hypotenuses of the triangles. We already know the length of $D_2Q$ and $D_1P$ , 9 and 6 (they're radii). So in order to find $PQ$ , we must find the length of the longer legs of the two triangles and add them.

$15^2 - 9^2 = (15-9)(15+9) = 6 \times 24 = 144$

$\sqrt{144} = 12$

$10^2-6^2 = (10-6)(10+6) = 4 \times 16 = 64$

$\sqrt{64} = 8$

Finally, the length of PQ is $12+8=\boxed{20}$ , or (C).

Solution 2

Using the above diagram, imagine that segment $\overline{QD_2}$ is shifted to the right to match up with $\overline{PD_1}$ . Then shift $\overline{QP}$ downwards to make a right triangle. We know $\overline{D_2D_1} = 25$ from the given information and the newly created leg has length $\overline{QD_2} + \overline{PD_1} = 9 + 6 = 15$ . Hence by Pythagorean theorem $15^2 + {\overline{QP}}^2 = 25^2$ .

$\overline{QP} = \boxed{20}$ , or C.

Video Solution

https://youtu.be/wVadnulGLok

@@ Line 40: / Line 40: @@
 label("$O$", (0,0), SW );
 </asy>
-Line PQ is tangent to both circles, so it forms a right angle with the radii (6 and 9). This, as well as the two vertical angles near O, prove triangles S<math>_2</math>QO and S<math>_1</math>PO similar by AA, with a scale factor of 6:9, or 2:3. Next, we must subdivide the line D<math>_2</math>D<math>_1</math> in a 2:3 ratio to get the length of the segments D<math>_2</math>O and D<math>_1</math>O. The total length is 10 - (-15), or 25, so applying the ratio, D<math>_2</math>O = '''15''' and D<math>_1</math>O = '''10'''. These are the hypotenuses of the triangles. We already know the length of D<math>_2</math>Q and D<math>_1</math>P, '''9''' and '''6''' (they're radii). So in order to find PQ, we must find the length of the longer legs of the two triangles and add them.
+Line PQ is tangent to both circles, so it forms a right angle with the radii (6 and 9). This, as well as the two vertical angles near O, prove triangles <math>D_2QO</math> and <math>D_1PO</math> similar by AA, with a scale factor of <math>6:9</math>, or <math>2:3</math>. Next, we must subdivide the line <math>D_2D_1</math> in a 2:3 ratio to get the length of the segments <math>D_2O</math> and <math>D_1O</math>. The total length is <math>10 - (-15)</math>, or <math>25</math>, so applying the ratio, <math>D_2O</math> = '''15''' and <math>D_1O</math> = '''10'''. These are the hypotenuses of the triangles. We already know the length of <math>D_2Q</math> and <math>D_1P</math>, '''9''' and '''6''' (they're radii). So in order to find <math>PQ</math>, we must find the length of the longer legs of the two triangles and add them.
 <math>15^2 - 9^2 = (15-9)(15+9) = 6 \times 24 = 144</math>
@@ Line 53: / Line 53: @@
 == Solution 2 ==
-Using the above diagram, imagine that segment <math>\overline{QS_2}</math> is shifted to the right to match up with <math>\overline{PS_1}</math>. Then shift <math>\overline{QP}</math> downwards to make a right triangle. We know <math>\overline{S_2S_1} = 25</math> from the given information and the newly created leg has length <math> \overline{QS_2} + \overline{PS_1} = 9 + 6 = 15</math>. Hence by Pythagorean theorem <math>15^2 + {\overline{QP}}^2 = 25^2</math>.
+Using the above diagram, imagine that segment <math>\overline{QD_2}</math> is shifted to the right to match up with <math>\overline{PD_1}</math>. Then shift <math>\overline{QP}</math> downwards to make a right triangle. We know <math>\overline{D_2D_1} = 25</math> from the given information and the newly created leg has length <math> \overline{QD_2} + \overline{PD_1} = 9 + 6 = 15</math>. Hence by Pythagorean theorem <math>15^2 + {\overline{QP}}^2 = 25^2</math>.
 <math> \overline{QP} = \boxed{20}</math>, or C.

2002 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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
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