Difference between revisions of "2021 AMC 12A Problems/Problem 17"
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Angle chasing reveals that <math>\triangle BPC\sim\triangle BDA</math>, therefore | Angle chasing reveals that <math>\triangle BPC\sim\triangle BDA</math>, therefore | ||
− | <cmath>2=\frac{BD}{BP}=\frac{AB}{BC}=\frac{AB}{43}</cmath> | + | <cmath>2=\frac{BD}{BP}=\frac{AB}{BC}=\frac{AB}{43},</cmath> |
− | < | + | or <math>AB=86</math>. |
+ | |||
Additional angle chasing shows that <math>\triangle ABO \sim\triangle CDO</math>, therefore | Additional angle chasing shows that <math>\triangle ABO \sim\triangle CDO</math>, therefore | ||
− | <cmath>2=\frac{AB}{CD}=\frac{BO}{OD}=\frac{BP+11}{BP-11}</cmath> | + | <cmath>2=\frac{AB}{CD}=\frac{BO}{OD}=\frac{BP+11}{BP-11},</cmath> |
− | < | + | or <math>BP=33</math> and <math>BD=66</math>. |
+ | |||
Since <math>\triangle ADB</math> is right, the Pythagorean theorem implies that | Since <math>\triangle ADB</math> is right, the Pythagorean theorem implies that | ||
− | <cmath>AD=\sqrt{86^2-66^2} | + | <cmath>AD=\sqrt{86^2-66^2}=4\sqrt{190}.</cmath> |
− | + | The answer is <math>4+190=\boxed{\textbf{(D) }194}</math>. | |
− | <math> | ||
~mn28407 (minor edits by eagleye) | ~mn28407 (minor edits by eagleye) | ||
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(2) Applying Pythagorean Theorem on <math>\triangle CDP</math> gives <math>(11+x)^2 + y^2 = 43^2.</math> | (2) Applying Pythagorean Theorem on <math>\triangle CDP</math> gives <math>(11+x)^2 + y^2 = 43^2.</math> | ||
− | (3) <math>\triangle BPC \sim \triangle BDA</math> with ratio <math>1:2,</math> so <math>AD = 2y | + | (3) <math>\triangle BPC \sim \triangle BDA</math> with ratio <math>1:2,</math> so <math>AD = 2y</math> using the fact that <math>P</math> is the midpoint of <math>BD</math>. |
− | Thus, <math>\frac{xt}{11} = 2y,</math> or <math>x = 22.</math> And <math>y = \sqrt{43^2 - 33^2} = 2 \sqrt{190},</math> so <math>AD = 4 \sqrt{190}</math> and the answer is <math>\boxed{194}.</math> | + | Thus, <math>\frac{xt}{11} = 2y,</math> or <math>x = 22.</math> And <math>y = \sqrt{43^2 - 33^2} = 2 \sqrt{190},</math> so <math>AD = 4 \sqrt{190}</math> and the answer is <math>4+190=\boxed{\textbf{(D) }194}.</math> |
~ ccx09 | ~ ccx09 | ||
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Therefore, <math>\overline{AC}</math> and <math>\overline{BD}</math> are both medians of <math>\triangle ABE</math>. This means that <math>O</math> is the centroid of <math>\triangle ABE</math>; therefore, because the centroid divides the median in a 2:1 ratio, <math>\frac{BO}{2} = DO = \frac{BD}{3}</math>. Recall that <math>P</math> is the midpoint of <math>BD</math>; <math>DP = \frac{BD}{2}</math>. The question tells us that <math>OP = 11</math>; <math>DP-DO=11</math>; we can write this in terms of <math>DB</math>; <math>\frac{DB}{2}-\frac{DB}{3} = \frac{DB}{6} = 11 \implies DB = 66</math>. | Therefore, <math>\overline{AC}</math> and <math>\overline{BD}</math> are both medians of <math>\triangle ABE</math>. This means that <math>O</math> is the centroid of <math>\triangle ABE</math>; therefore, because the centroid divides the median in a 2:1 ratio, <math>\frac{BO}{2} = DO = \frac{BD}{3}</math>. Recall that <math>P</math> is the midpoint of <math>BD</math>; <math>DP = \frac{BD}{2}</math>. The question tells us that <math>OP = 11</math>; <math>DP-DO=11</math>; we can write this in terms of <math>DB</math>; <math>\frac{DB}{2}-\frac{DB}{3} = \frac{DB}{6} = 11 \implies DB = 66</math>. | ||
− | + | We are almost finished. Each side length of <math>\triangle ABD</math> is twice as long as the corresponding side length <math>\triangle CBP</math> or <math>\triangle CPD</math>, since those triangles are similar; this means that <math>AB = 2 \cdot 43 = 86</math>. Now, by Pythagorean theorem on <math>\triangle ABD</math>, <math>AB^{2} - BD^{2} = AD^{2} \implies 86^{2}-66^{2} = AD^{2} \implies AD = \sqrt{3040} \implies AD = 4 \sqrt{190}</math>. <math>4+190 = \boxed{\textbf{(D) }194}</math> | |
− | We are almost finished. Each side length of <math>\triangle ABD</math> is twice as long as the corresponding side length <math>\triangle CBP</math> or <math>\triangle CPD</math>, since those triangles are similar; this means that <math>AB = 2 \cdot 43 = 86</math>. Now, by Pythagorean theorem on <math>\triangle ABD</math>, <math>AB^{2} - BD^{2} = AD^{2} \implies 86^{2}-66^{2} = AD^{2} \implies AD = \sqrt{3040} \implies AD = 4 \sqrt{190}</math>. <math>4+190 = \boxed{ | ||
~ ihatemath123 | ~ ihatemath123 |
Revision as of 15:11, 22 August 2021
- The following problem is from both the 2021 AMC 10A #17 and 2021 AMC 12A #17, so both problems redirect to this page.
Contents
- 1 Problem
- 2 Diagram
- 3 Solution 1
- 4 Solution 2 (Similar Triangles, Areas, Pythagorean Theorem)
- 5 Solution 3 (Short)
- 6 Solution 4 (Extending the Line)
- 7 Video Solution (Using Similar Triangles, Pythagorean Theorem)
- 8 Video Solution by Punxsutawney Phil
- 9 Video Solution by Mathematical Dexterity
- 10 See also
Problem
Trapezoid has
, and
. Let
be the intersection of the diagonals
and
, and let
be the midpoint of
. Given that
, the length of
can be written in the form
, where
and
are positive integers and
is not divisible by the square of any prime. What is
?
Diagram
~MRENTHUSIASM (by Geometry Expressions)
Solution 1
Angle chasing reveals that , therefore
or
.
Additional angle chasing shows that , therefore
or
and
.
Since is right, the Pythagorean theorem implies that
The answer is
.
~mn28407 (minor edits by eagleye)
Solution 2 (Similar Triangles, Areas, Pythagorean Theorem)
Since is isosceles with base
it follows that median
is also an altitude. Let
and
so
Since by vertical angles, we conclude that
by AA, from which
or
Let the brackets denote areas. Notice that
(By the same base and height, we deduce that
Subtracting
from both sides gives
). Doubling both sides produces
Rearranging and factoring result in
from which
Applying the Pythagorean Theorem to right we have
Finally, we get
so the answer is
~MRENTHUSIASM
Solution 3 (Short)
Let .
a is perpendicular bisector of
Then, let
thus
(1) so we get
or
(2) Applying Pythagorean Theorem on gives
(3) with ratio
so
using the fact that
is the midpoint of
.
Thus, or
And
so
and the answer is
~ ccx09
Solution 4 (Extending the Line)
Observe that is congruent to
; both are similar to
. Let's extend
and
past points
and
respectively, such that they intersect at a point
. Observe that
is
degrees, and that
. Thus, by ASA, we know that
, thus,
, meaning
is the midpoint of
.
Let
be the midpoint of
. Note that
is congruent to
, thus
, meaning
is the midpoint of
Therefore, and
are both medians of
. This means that
is the centroid of
; therefore, because the centroid divides the median in a 2:1 ratio,
. Recall that
is the midpoint of
;
. The question tells us that
;
; we can write this in terms of
;
.
We are almost finished. Each side length of is twice as long as the corresponding side length
or
, since those triangles are similar; this means that
. Now, by Pythagorean theorem on
,
.
~ ihatemath123
Video Solution (Using Similar Triangles, Pythagorean Theorem)
~ pi_is_3.14
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=rtdovluzgQs
Video Solution by Mathematical Dexterity
https://www.youtube.com/watch?v=QzAVdsgBBqg
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 |
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.