Difference between revisions of "2000 AMC 12 Problems/Problem 19"
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<math>\text {(A)}\ 2 \qquad \text {(B)}\ 2.5 \qquad \text {(C)}\ 3 \qquad \text {(D)}\ 3.5 \qquad \text {(E)}\ 4</math> | <math>\text {(A)}\ 2 \qquad \text {(B)}\ 2.5 \qquad \text {(C)}\ 3 \qquad \text {(D)}\ 3.5 \qquad \text {(E)}\ 4</math> | ||
− | == Solution == | + | == Solution 1== |
The answer is exactly <math>3</math>, choice <math>\mathrm{(C)}</math>. | The answer is exactly <math>3</math>, choice <math>\mathrm{(C)}</math>. | ||
− | We can find the area of triangle <math>ADE</math> by using the simple formula <math>\frac{bh}{2}</math>. Dropping an altitude from <math>A</math>, we see that it has length <math>12</math> ( we can split the large triangle into a <math>9-12-15</math> and a <math>5-12-13</math> triangle). Then we can apply the Angle Bisector | + | We can find the area of triangle <math>ADE</math> by using the simple formula <math>\frac{bh}{2}</math>. Dropping an altitude from <math>A</math>, we see that it has length <math>12</math> (we can split the large triangle into a <math>9-12-15</math> and a <math>5-12-13</math> triangle). Then we can apply the [[Angle Bisector Theorem]] on triangle <math>ABC</math> to solve for <math>BE</math>. Solving <math>\frac{13}{BE}=\frac{15}{14-BE}</math>, we get that <math>BE=\frac{13}{2}</math>. <math>D</math> is the midpoint of <math>BC</math> so <math>BD=7</math>. Thus we get the base of triangle <math>ADE, DE</math>, to be <math>\frac{1}{2}</math> units long. Applying the formula <math>\frac{bh}{2}</math>, we get <math>\frac{12*\frac{1}{2}}{2}=3</math>. |
+ | |||
+ | ==Solution 2== | ||
+ | The area of <math>ADE</math> is <math>\frac{DE\cdot h}{2}=\frac{DE}{BC} \cdot \frac{BC\cdot h}{2}=\frac{DE}{BC}[ABC]</math> where <math>h</math> is the height of triangle <math>ABC</math>. Using Angle Bisector Theorem, we find <math>\frac{13}{BE}=\frac{15}{14-BE}</math>, which we solve to get <math>BE=\frac{13}{2}</math>. <math>D</math> is the midpoint of <math>BC</math> so <math>BD=7</math>. Thus we get the base of triangle <math>ADE, DE</math>, to be <math>\frac{1}{2}</math> units long. We can now use [[Heron's Formula]] on <math>ABC</math>. | ||
+ | <cmath>s=\frac{AB+BC+AC}{2}=21</cmath> | ||
+ | <cmath>[ABC]=\sqrt{(s)(s-AB)(s-BC)(s-AC)}=\sqrt{(21)(8)(7)(6)}=84</cmath> | ||
+ | <cmath>\frac{DE}{BC}[ABC]=\frac{\frac{1}{2}}{14}\cdot 84=3</cmath> | ||
+ | Therefore, the answer is <math>\mathrm{C}</math>. | ||
+ | |||
+ | ==Solution 3== | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D,E; | ||
+ | B=(0,0); | ||
+ | C=(14,0); | ||
+ | A=intersectionpoint(arc(B,13,0,90),arc(C,15,90,180)); | ||
+ | draw(A--B--C--cycle); | ||
+ | D=(7,0); | ||
+ | E=(6.5,0); | ||
+ | draw(A--E); | ||
+ | draw(A--D); | ||
+ | label("$A$",A,N); | ||
+ | label("$B$",B,S); | ||
+ | label("$C$",C,S); | ||
+ | label("$E$",E,NW); | ||
+ | label("$D$",D,NE); | ||
+ | label("$13$",A--B,NW); | ||
+ | label("$15$",A--C,NE); | ||
+ | label("$14$",B--C,S); | ||
+ | label("$6.5$",B--E,N); | ||
+ | label("$7$",C--D,N); | ||
+ | </asy> | ||
+ | |||
+ | Let's find the area of <math>\Delta ABC</math> by Heron, | ||
+ | |||
+ | <math>s=\frac{a+b+c}{2}\\\\s=\frac{14+15+13}{2}\to\boxed{s=21}</math> | ||
+ | |||
+ | Then, | ||
+ | |||
+ | <math>A^2=s(s-a)(s-b)(s-c)\\\\A^2=21(21-14)(21-15)(21-13)\\\\\boxed{A=84}</math> | ||
+ | |||
+ | Knowing that D is the midpoint of BC, then <math>BD=CD=7</math>. | ||
+ | |||
+ | By [[Angle Bisector Theorem]] we know that: | ||
+ | |||
+ | <math>\frac{13}{BE}=\frac{15}{CE}</math> | ||
+ | |||
+ | <math>\boxed{CE=14-BE}</math> | ||
+ | |||
+ | <math>\frac{13}{BE}=\frac{15}{14-BE}</math> | ||
+ | |||
+ | <math>\boxed{BE=6.5}\Rightarrow CE=7.5</math> | ||
+ | |||
+ | Also, we know that: | ||
+ | |||
+ | <math>\frac{A_{\Delta ADE}}{A_{\Delta ABC}}=\frac{ED}{BC}</math> | ||
+ | |||
+ | And, we can easily see that <math>DE=0.5</math>, so, | ||
+ | |||
+ | $\frac{A_{ | ||
+ | |||
+ | == Video Solution == | ||
+ | https://youtu.be/A5oxZp3jem4 | ||
== See also == | == See also == |
Latest revision as of 17:37, 17 February 2024
Problem
In triangle ,
,
,
. Let
denote the midpoint of
and let
denote the intersection of
with the bisector of angle
. Which of the following is closest to the area of the triangle
?
Solution 1
The answer is exactly , choice
.
We can find the area of triangle
by using the simple formula
. Dropping an altitude from
, we see that it has length
(we can split the large triangle into a
and a
triangle). Then we can apply the Angle Bisector Theorem on triangle
to solve for
. Solving
, we get that
.
is the midpoint of
so
. Thus we get the base of triangle
, to be
units long. Applying the formula
, we get
.
Solution 2
The area of is
where
is the height of triangle
. Using Angle Bisector Theorem, we find
, which we solve to get
.
is the midpoint of
so
. Thus we get the base of triangle
, to be
units long. We can now use Heron's Formula on
.
Therefore, the answer is
.
Solution 3
Let's find the area of by Heron,
Then,
Knowing that D is the midpoint of BC, then .
By Angle Bisector Theorem we know that:
Also, we know that:
And, we can easily see that , so,
$\frac{A_{
Video Solution
See also
2000 AMC 12 (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 | |
All AMC 12 Problems and Solutions |
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