Difference between revisions of "2004 AMC 12A Problems/Problem 8"
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=== Solution 5 === | === Solution 5 === | ||
− | We want to figure out <math> | + | We want to figure out <math>[\triangle ADE] - [\triangle BDC]</math>. |
Notice that <math>\triangle ABC</math> and <math>\triangle BAE</math> "intersect" and form <math>\triangle ADB</math>. | Notice that <math>\triangle ABC</math> and <math>\triangle BAE</math> "intersect" and form <math>\triangle ADB</math>. | ||
− | This means that <math> | + | This means that <math>[\triangle BAE] - [\triangle ABC)] = [\triangle ADE] - [\triangle BDC]</math> because <math>[\triangle ADB]</math> cancels out, which can be seen easily in the diagram. |
− | <math> | + | <math>[\triangle BAE] = 0.5 \cdot 4 \cdot 8 = 16</math> |
− | <math> | + | <math>[\triangle ABC] = 0.5 \cdot 4 \cdot 16 = 12</math> |
− | <math> | + | <math>[\triangle BDC] - [\triangle ADE] = 16 - 12 =\boxed{\mathrm{(B)}\ 4}</math> |
==Video Solution== | ==Video Solution== |
Revision as of 23:24, 28 September 2024
- The following problem is from both the 2004 AMC 12A #8 and 2004 AMC 10A #9, so both problems redirect to this page.
Contents
Problem
In the overlapping triangles and
sharing common side
,
and
are right angles,
,
,
, and
and
intersect at
. What is the difference between the areas of
and
?
Solutions
Solution 1
Looking, we see that the area of is 16 and the area of
is 12. Set the area of
to be x. We want to find
-
. So, that would be
and
. Therefore,
~ MathKatana
Solution 2
Since and
,
. By alternate interior angles and
, we find that
, with side length ratio
. Their heights also have the same ratio, and since the two heights add up to
, we have that
and
. Subtracting the areas,
.
Solution 3
Let represent the area of figure
. Note that
and
.
Solution 4 (coordbash)
Put figure on a graph.
goes from (0, 0) to (4, 6) and
goes from (4, 0) to (0, 8).
is on line
.
is on line
. Finding intersection between these points,
.
This gives us the x-coordinate of D.
So, is the height of
, then area of
is
Now, the height of is
And the area of
is
This gives us
Therefore, the difference is
Solution 5
We want to figure out .
Notice that
and
"intersect" and form
.
This means that because
cancels out, which can be seen easily in the diagram.
Video Solution
Education, the Study of Everything
See also
2004 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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 |
2004 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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 10 Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.