Difference between revisions of "2001 AMC 10 Problems/Problem 24"
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Solution By: MATHCOUNTSCMS25 | Solution By: MATHCOUNTSCMS25 | ||
− | + | Fixed <math>\text{\LaTeX}</math> - Mliu630XYZ, palaashgang, anshulb, JoyfulSapling | |
− | + | ==Solution 4 (EZ Cheez) == | |
− | + | Choose any value for <math>BC</math>, and then use Pythagorean theorem to get <math>CD - AB</math>, and <math>AB = (BC - (CD - AB))/2</math>). Then multiply <math>AB \cdot CD</math>. | |
− | + | For example: | |
+ | |||
+ | <math>BC=25</math>. <math>CD - AB = \sqrt{25^2 - 7^2} = 24</math>. <math>AB = (25 - 24)/2=0.5</math>. <math>CD = 0.5 + 24 = 24.5</math>. | ||
+ | <math>AB \cdot CD = (0.5)(24.5)= \boxed{\textbf{(B)}\ 12.25}</math>. | ||
==See Also== | ==See Also== |
Latest revision as of 17:58, 3 March 2024
Problem
In trapezoid ,
and
are perpendicular to
, with
,
, and
. What is
?
Solution
If and
, then
. By the Pythagorean theorem, we have
Solving the equation, we get
.
Solution 2
Simpler is just drawing the trapezoid and then using what is given to solve.
Draw a line parallel to that connects the longer side to the corner of the shorter side. Name the bottom part
and top part
.
By the Pythagorean theorem, it is obvious that
(the RHS is the fact the two sides added together equals that). Then, we get
, cancel out and factor and we get
. Notice that
is what the question asks, so the answer is
.
Solution by IronicNinja
Solution 3
We know it is a trapezoid and that and
are perpendicular to
. If they are perpendicular to
that means this is a right-angle trapezoid (search it up if you don't know what it looks like or you can look at the trapezoid in the first solution). We know
is
. We can then set the length of
to be
and the length of
to be
.
would then be
. Let's draw a straight line down from point
which is perpendicular to
and parallel to
. Let's name this line
. Then let's name the point at which line
intersects
point
. Line
partitions the trapezoid into ▭
and
. We will use the triangle to solve for
using the Pythagorean theorem. The line segment
would be
because
is
and
is
.
is
because it is parallel to
and both are of equal length. Because of the Pythagorean theorem, we know that
. Substituting the values we have we get
. Simplifying this we get
. Now we get rid of the
and
terms from both sides to get
. Combining like terms we get
. Then we divide by
to get
. Now we know that
which is answer choice
.
Solution By: MATHCOUNTSCMS25
Fixed - Mliu630XYZ, palaashgang, anshulb, JoyfulSapling
Solution 4 (EZ Cheez)
Choose any value for , and then use Pythagorean theorem to get
, and
). Then multiply
.
For example:
.
.
.
.
.
See Also
2001 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.