Difference between revisions of "2000 AMC 12 Problems/Problem 19"
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== Solution 1== | == 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 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>. | + | 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== | ==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>. | + | 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>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>[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> | <cmath>\frac{DE}{BC}[ABC]=\frac{\frac{1}{2}}{14}\cdot 84=3</cmath> | ||
Therefore, the answer is <math>\mathrm{C}</math>. | Therefore, the answer is <math>\mathrm{C}</math>. | ||
− | |||
==Solution 3== | ==Solution 3== | ||
<asy> | <asy> | ||
− | |||
pair A,B,C,D,E; | pair A,B,C,D,E; | ||
B=(0,0); | B=(0,0); | ||
Line 37: | Line 35: | ||
label("$15$",A--C,NE); | label("$15$",A--C,NE); | ||
label("$14$",B--C,S); | label("$14$",B--C,S); | ||
+ | label("$6.5$",B--E,N); | ||
+ | label("$7$",C--D,N); | ||
</asy> | </asy> | ||
Line 47: | Line 47: | ||
<math>A^2=s(s-a)(s-b)(s-c)\\\\A^2=21(21-14)(21-15)(21-13)\\\\\boxed{A=84}</math> | <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, | + | Knowing that D is the midpoint of BC, then <math>BD=CD=7</math>. |
By [[Angle Bisector Theorem]] we know that: | By [[Angle Bisector Theorem]] we know that: | ||
Line 65: | Line 65: | ||
And, we can easily see that <math>DE=0.5</math>, so, | 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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