Difference between revisions of "2018 AMC 10A Problems/Problem 13"
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== Solution 2 (Draw it, not recommended) == | == Solution 2 (Draw it, not recommended) == | ||
− | Use the ruler and graph paper you brought to quickly draw a 3-4-5 triangle of any scale (don't trust the diagram in the booklet). Very carefully fold the acute vertices together and make a crease. Measure the crease with the ruler. If you were reasonably careful, you should see that it measures somewhat more than <math>\frac{7}{4}</math> units and somewhat less than <math>2</math> units. The only answer choice in that range is <math>\boxed{\textbf{ (D) } \frac{15}{8}}</math>. | + | Use the ruler and graph paper you brought to quickly draw a 3-4-5 triangle of any scale (don't trust the diagram in the booklet). Very carefully fold the acute vertices together and make a crease. Measure the crease with the ruler. If you were reasonably careful, you should see that it measures somewhat more than <math>\frac{7}{4}</math> units and somewhat less than <math>2</math> units. The only answer choice in that range is <math>\boxed{\textbf{(D) } \frac{15}{8}}</math>. |
This is pretty much a cop-out, but it's allowed in the rules technically. This is basically useless for problems without diagrams. | This is pretty much a cop-out, but it's allowed in the rules technically. This is basically useless for problems without diagrams. | ||
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<cmath>\frac{3}{2}^2+\frac{9}{8}^2=DF^2 \Rightarrow \frac{9}{4}+\frac{9\cdot9}{8\cdot8}=DF^2 \Rightarrow \frac{9\cdot2\cdot8}{4\cdot2\cdot8}+\frac{9\cdot9}{8\cdot8}=DF^2 \Rightarrow \frac{9\cdot2\cdot8+9\cdot9}{8\cdot8}=DF^2 \Rightarrow \frac{9(2\cdot8+9)}{8\cdot8}=DF^2</cmath> | <cmath>\frac{3}{2}^2+\frac{9}{8}^2=DF^2 \Rightarrow \frac{9}{4}+\frac{9\cdot9}{8\cdot8}=DF^2 \Rightarrow \frac{9\cdot2\cdot8}{4\cdot2\cdot8}+\frac{9\cdot9}{8\cdot8}=DF^2 \Rightarrow \frac{9\cdot2\cdot8+9\cdot9}{8\cdot8}=DF^2 \Rightarrow \frac{9(2\cdot8+9)}{8\cdot8}=DF^2</cmath> | ||
<cmath>\Rightarrow DF=\sqrt{\frac{9(2\cdot8+9)}{8\cdot8}} \Rightarrow DF=\frac{3\cdot5}{8} \Rightarrow DF=\frac{15}{8}</cmath> | <cmath>\Rightarrow DF=\sqrt{\frac{9(2\cdot8+9)}{8\cdot8}} \Rightarrow DF=\frac{3\cdot5}{8} \Rightarrow DF=\frac{15}{8}</cmath> | ||
− | Thus, our answer is <math>\boxed{\textbf{D) } \frac{15}{8}}</math>. | + | Thus, our answer is <math>\boxed{\textbf{(D) } \frac{15}{8}}</math>. |
+ | |||
+ | ~ (minor edits) <B+ | ||
== Solution 4 == | == Solution 4 == | ||
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By the Distance Formula, | By the Distance Formula, | ||
− | <math>(2-\frac{7}{8})^2+({\frac{3}{2}})^2=d^2 \implies d^2=\frac{225}{64} \implies d=\boxed{\frac{15}{8}}</math> | + | <math>(2-\frac{7}{8})^2+({\frac{3}{2}})^2=d^2 \implies d^2=\frac{225}{64} \implies d=\boxed{\textbf{(D) }\frac{15}{8}}</math> |
== Solution 6 == | == Solution 6 == | ||
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The area of <math>\triangle{BCD} \text{ is thus} = \frac{7}{8}*3*\frac{1}{2} \implies \frac{21}{16}.</math> Because <math>[ADB]=[ABC]-[BCD],</math> we can plug in and see that <math>[ADB]=6-\frac{21}{16}=\frac{75}{16}.</math> | The area of <math>\triangle{BCD} \text{ is thus} = \frac{7}{8}*3*\frac{1}{2} \implies \frac{21}{16}.</math> Because <math>[ADB]=[ABC]-[BCD],</math> we can plug in and see that <math>[ADB]=6-\frac{21}{16}=\frac{75}{16}.</math> | ||
− | Since <math>A=\frac{bh}{2}</math>, we get <math>\frac{75}{16}=\frac{5h}{2} \implies 5h=\frac{75}{8} \implies h=\boxed{\frac{15}{8}}</math> | + | Since <math>A=\frac{bh}{2}</math>, we get <math>\frac{75}{16}=\frac{5h}{2} \implies 5h=\frac{75}{8} \implies h=\boxed{\textbf{(D) }\frac{15}{8}}</math> |
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==Video Solution (HOW TO THINK CREATIVELY!)== | ==Video Solution (HOW TO THINK CREATIVELY!)== |
Latest revision as of 11:39, 18 August 2024
- The following problem is from both the 2018 AMC 10A #13 and 2018 AMC 12A #11, so both problems redirect to this page.
Contents
Problem
A paper triangle with sides of lengths and
inches, as shown, is folded so that point
falls on point
. What is the length in inches of the crease?
Solution 1
First, we need to realize that the crease line is just the perpendicular bisector of side , the hypotenuse of right triangle
. Call the midpoint of
point
. Draw this line and call the intersection point with
as
. Now,
is similar to
by
similarity. Setting up the ratios, we find that
Thus, our answer is
.
Note
In general, whenever we are asked to make a crease, think about that crease as a line of reflection over which the diagram is reflected. This is why the crease must be the perpendicular bisector of , because
must be reflected onto
.
Solution 2 (Draw it, not recommended)
Use the ruler and graph paper you brought to quickly draw a 3-4-5 triangle of any scale (don't trust the diagram in the booklet). Very carefully fold the acute vertices together and make a crease. Measure the crease with the ruler. If you were reasonably careful, you should see that it measures somewhat more than units and somewhat less than
units. The only answer choice in that range is
.
This is pretty much a cop-out, but it's allowed in the rules technically. This is basically useless for problems without diagrams.
~ (minor edits) <B+
Solution 3
Since is a right triangle, we can see that the slope of line
is
=
. We know that if we fold
so that point
meets point
the crease line will be perpendicular to
and we also know that the slopes of perpendicular lines are negative reciprocals of one another. Then, we can see that the slope of our crease line is
.
Let us call the midpoint of point
, the midpoint of
point
, and the crease line
. We know that
is parallel to
and that
's length is
. Using our slope calculation from earlier, we can see that
. With this information, we can solve for
:
We can then use the Pythagorean Theorem to find
.
Thus, our answer is
.
~ (minor edits) <B+
Solution 4
Make use of the diagram in Solution 3. It can be deduced that . Let
. In
,
. Then
also would be
.
In ,
. After some quick math, we get
.
Solving for
will give
.
.
Solution 5
Like in Solution 3, we can make use of coordinate geometry to solve this problem. Because the length of the crease is constant no matter the positioning of the triangle, reorient the triangle so it has C at the origin and B and A at and
respectively.
We know that each point in the crease is equidistant from and
, so the crease must pass through the midpoint of
, which is
, and be perpendicular to hypotenuse
. The crease therefore has a slope of
.
Plugging into point-slope we find that the equation of the crease is,
Using 0 for y, we see that the crease intersects at
.
By the Distance Formula,
Solution 6
Let be a point on AC such that
and
is isosceles. The altitude of this triangle is the crease of the paper triangle.
Setting , we have a right triangle
with legs of x and 3, and, because
hypotenuse of
.
The area of Because
we can plug in and see that
Since , we get
Video Solution (HOW TO THINK CREATIVELY!)
~Education, the Study of Everything
Video Solution
~ Whiz
Video Solution by OmegaLearn
https://youtu.be/4_x1sgcQCp4?t=2623
~ pi_is_3.14
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 |
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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.