Difference between revisions of "1989 AIME Problems/Problem 15"
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− | Because we're given three concurrent [[cevian]]s and their lengths, it seems very tempting to apply [[mass point]]s. We immediately see that <math>w_E = 3</math>, <math>w_B = 1</math>, and <math>w_A = w_D = 2</math>. Now, we recall that the masses on the three sides of the triangle must be balanced out, so <math>w_C = 1</math> and <math> | + | Because we're given three concurrent [[cevian]]s and their lengths, it seems very tempting to apply [[mass point]]s. We immediately see that <math>w_E = 3</math>, <math>w_B = 1</math>, and <math>w_A = w_D = 2</math>. Now, we recall that the masses on the three sides of the triangle must be balanced out, so <math>w_C = 1</math> and <math>w_F = 3</math>. Thus, <math>CP = 15</math> and <math>PF = 5</math>. |
Recalling that <math>w_C = w_B = 1</math>, we see that <math>DC = DB</math> and <math>DP</math> is a [[median]] to <math>BC</math> in <math>\triangle BCP</math>. Applying [[Stewart's Theorem]], <math>BC^2 + 12^2 = 2(15^2 + 9^2)</math>, and <math>BC = 6\sqrt {13}</math>. Now notice that <math>2[BCP] = [ABC]</math>, because both triangle share the same base and the <math>h_{\triangle ABC} = 2h_{\triangle BCP}</math>. Applying [[Heron's formula]] on triangle <math>BCP</math> with sides <math>15</math>, <math>9</math>, and <math>6\sqrt3</math>, <math>[BCP] = 54</math> and <math>[ABC] = \boxed{108}</math>. | Recalling that <math>w_C = w_B = 1</math>, we see that <math>DC = DB</math> and <math>DP</math> is a [[median]] to <math>BC</math> in <math>\triangle BCP</math>. Applying [[Stewart's Theorem]], <math>BC^2 + 12^2 = 2(15^2 + 9^2)</math>, and <math>BC = 6\sqrt {13}</math>. Now notice that <math>2[BCP] = [ABC]</math>, because both triangle share the same base and the <math>h_{\triangle ABC} = 2h_{\triangle BCP}</math>. Applying [[Heron's formula]] on triangle <math>BCP</math> with sides <math>15</math>, <math>9</math>, and <math>6\sqrt3</math>, <math>[BCP] = 54</math> and <math>[ABC] = \boxed{108}</math>. |
Revision as of 21:10, 26 January 2008
Problem
Point is inside . Line segments , , and are drawn with on , on , and on (see the figure below). Given that , , , , and , find the area of .
Solution
Solution 1
Because we're given three concurrent cevians and their lengths, it seems very tempting to apply mass points. We immediately see that , , and . Now, we recall that the masses on the three sides of the triangle must be balanced out, so and . Thus, and .
Recalling that , we see that and is a median to in . Applying Stewart's Theorem, , and . Now notice that , because both triangle share the same base and the . Applying Heron's formula on triangle with sides , , and , and .
Solution 2
Using a different form of Ceva's Theorem, we have
Solving and , we obtain and .
Let be the point on such that . Since and , . (Midline Theorem)
Also, since and $\QD = \frac12FC$ (Error compiling LaTeX. Unknown error_msg), we see that , , etc. (Midline Theorem) Similarly, we have () and thus .
is a right triangle, so () is . Therefore, the area of . Using area ratio, .
Solution 3
Because the length of cevian is unknown, we can examine what happens when we extend it or decrease its length and see that it simply changes the angles between the cevians. Wouldn't it be great if it the length of was such that ? Let's first assume it's a right angle and hope that everything works out.
Extend to so that . The result is that , , and because . Now we see that if we are able to show that , that is , then our right angle assumption will be true.
Apply the Pythagorean Theorem on to get , so and . Now, we apply the Law of Cosines on triangles and .
Let . Notice that and , so we get two nice equations.
Solving, (yay!).
Now, the area is easy to find. .
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Final Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |