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Difference between revisions of "2021 AIME II Problems/Problem 5"

m (→‎Solution 2 (Casework: Detailed Explanation of Solution 1))
m (→‎Solution 2 (Casework: Detailed Explanation of Solution 1))
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For one such obtuse triangle, let <math>4,10,</math> and <math>x</math> be its side-lengths and <math>K</math> be its area. By the Pythagorean Inequality Theorem, it is clear that <math>x\neq10.</math> We will use casework on its longest side:
 
For one such obtuse triangle, let <math>4,10,</math> and <math>x</math> be its side-lengths and <math>K</math> be its area. By the Pythagorean Inequality Theorem, it is clear that <math>x\neq10.</math> We will use casework on its longest side:
  
−
<b>Case (1): The longest side has length <math>\boldsymbol{10,}</math> so <math>\boldsymbol{0<x<10.}</math></b>
+
<b>Case (1): Its longest side has length <math>\boldsymbol{10,}</math> so <math>\boldsymbol{0<x<10.}</math></b>
  
 
By the Triangle Inequality Theorem, we have <math>4+x>10,</math> from which <math>x>6.</math>
 
By the Triangle Inequality Theorem, we have <math>4+x>10,</math> from which <math>x>6.</math>
Line 27: Line 27:
 
At <math>x=6,</math> the obtuse triangle degenerates into a straight line with area <math>K=0;</math> at <math>x=\sqrt{84},</math> the obtuse triangle degenerates into a right triangle with area <math>K=\frac12\cdot4\cdot\sqrt{84}=2\sqrt{84}.</math> Together, we obtain <math>0<K<2\sqrt{84},</math> or <math>K\in\left(0,2\sqrt{84}\right).</math>
 
At <math>x=6,</math> the obtuse triangle degenerates into a straight line with area <math>K=0;</math> at <math>x=\sqrt{84},</math> the obtuse triangle degenerates into a right triangle with area <math>K=\frac12\cdot4\cdot\sqrt{84}=2\sqrt{84}.</math> Together, we obtain <math>0<K<2\sqrt{84},</math> or <math>K\in\left(0,2\sqrt{84}\right).</math>
  
−
<b>Case (2): The longest side has length <math>\boldsymbol{x,}</math> so <math>\boldsymbol{x>10.}</math></b>
+
<b>Case (2): Its longest side has length <math>\boldsymbol{x,}</math> so <math>\boldsymbol{x>10.}</math></b>
  
 
By the Triangle Inequality Theorem, we have <math>4+10>x,</math> from which <math>x<14.</math>
 
By the Triangle Inequality Theorem, we have <math>4+10>x,</math> from which <math>x<14.</math>

Revision as of 13:32, 29 April 2021

Problem

For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.

Solution 1

We start by defining a triangle. The two small sides MUST add to a larger sum than the long side. We are given 4 and 10 as the sides, so we know that the 3rd side is between 6 and 14, exclusive. We also have to consider the word OBTUSE triangles. That means that the two small sides squared is less than the 3rd side. So the triangles sides are between 6 and $\sqrt{84}$ exclusive, and the larger bound is between $\sqrt{116}$ and 14, exclusive. The area of these triangles are from 0 (straight line) to $2\sqrt{84}$ on the first "small bound" and the larger bound is between 0 and 20. $0 < s < 2\sqrt{84}$ is our first equation, and $0 < s < 20$ is our 2nd equation. Therefore, the area is between $\sqrt{336}$ and $\sqrt{400}$, so our final answer is $\boxed{736}$.

~ARCTICTURN

Solution 2 (Casework: Detailed Explanation of Solution 1)

Every obtuse triangle must satisfy both of the following:

  • Triangle Inequality Theorem: If $a,b,$ and $c$ are the side-lengths of a triangle with $a\leq b\leq c,$ then $a+b>c.$
  • Pythagorean Inequality Theorem: If $a,b,$ and $c$ are the side-lengths of an obtuse triangle with $a\leq b<c,$ then $a^2+b^2<c^2.$

For one such obtuse triangle, let $4,10,$ and $x$ be its side-lengths and $K$ be its area. By the Pythagorean Inequality Theorem, it is clear that $x\neq10.$ We will use casework on its longest side:

Case (1): Its longest side has length $\boldsymbol{10,}$ so $\boldsymbol{0<x<10.}$

By the Triangle Inequality Theorem, we have $4+x>10,$ from which $x>6.$

By the Pythagorean Inequality Theorem, we have $4^2+x^2<10^2,$ from which $x<\sqrt{84}.$

Taking the intersection of these three inequalities produces $6<x<\sqrt{84}$ for this case.

At $x=6,$ the obtuse triangle degenerates into a straight line with area $K=0;$ at $x=\sqrt{84},$ the obtuse triangle degenerates into a right triangle with area $K=\frac12\cdot4\cdot\sqrt{84}=2\sqrt{84}.$ Together, we obtain $0<K<2\sqrt{84},$ or $K\in\left(0,2\sqrt{84}\right).$

Case (2): Its longest side has length $\boldsymbol{x,}$ so $\boldsymbol{x>10.}$

By the Triangle Inequality Theorem, we have $4+10>x,$ from which $x<14.$

By the Pythagorean Inequality Theorem, we have $4^2+10^2<x^2,$ from which $x>\sqrt{116}.$

Taking the intersection of these three inequalities produces $\sqrt{116}<x<14$ for this case.

At $x=14,$ the obtuse triangle degenerates into a straight line with area $K=0;$ at $x=\sqrt{116},$ the obtuse triangle degenerates into a right triangle with area $K=\frac12\cdot4\cdot10=20.$ Together, we obtain $0<K<20,$ or $K\in\left(0,20\right).$

Answer

It is possible for non-congruent obtuse triangles to have the same area. Therefore, all such positive real numbers $s$ are in exactly one of $\left(0,2\sqrt{84}\right)$ or $\left(0,20\right).$ Taking the exclusive disjunction, the set of all such $s$ is \[[a,b)=\left(0,2\sqrt{84}\right)\oplus\left(0,20\right)=\left[2\sqrt{84},20\right),\] from which $a^2+b^2=\boxed{736}.$

~MRENTHUSIASM

Solution 3

We have the diagram below.

[asy] draw((0,0)--(1,2*sqrt(3))); draw((1,2*sqrt(3))--(10,0)); draw((10,0)--(0,0)); label("A",(0,0),SW); label("B",(1,2*sqrt(3)),N); label("C",(10,0),SE); label("$\theta$",(0,0),NE); label("$\alpha$",(1,2*sqrt(3)),SSE); label("$4$",(0,0)--(1,2*sqrt(3)),WNW); label("$10$",(0,0)--(10,0),S); [/asy]

We proceed by taking cases on the angles that can be obtuse, and finding the ranges for $s$ that they yield .

If angle $\theta$ is obtuse, then we have that $s \in (0,20)$. This is because $s=20$ is attained at $\theta = 90^{\circ}$, and the area of the triangle is strictly decreasing as $\theta$ increases beyond $90^{\circ}$. This can be observed from \[s=\frac{1}{2}(4)(10)\sin\theta\]by noting that $\sin\theta$ is decreasing in $\theta \in (90^{\circ},180^{\circ})$.

Then, we note that if $\alpha$ is obtuse, we have $s \in (0,4\sqrt{21})$. This is because we get $x=\sqrt{10^2-4^2}=\sqrt{84}=2\sqrt{21}$ when $\alpha=90^{\circ}$, yileding $s=4\sqrt{21}$. Then, $s$ is decreasing as $\alpha$ increases by the same argument as before.

$\angle{ACB}$ cannot be obtuse since $AC>AB$.

Now we have the intervals $s \in (0,20)$ and $s \in (0,4\sqrt{21})$ for the cases where $\theta$ and $\alpha$ are obtuse, respectively. We are looking for the $s$ that are in exactly one of these intervals, and because $4\sqrt{21}<20$, the desired range is \[s\in [4\sqrt{21},20)\]giving \[a^2+b^2=\boxed{736}\Box\]

Solution 4

Note: Archimedes15 Solution which I added an answer here are two cases. Either the $4$ and $10$ are around an obtuse angle or the $4$ and $10$ are around an acute triangle. If they are around the obtuse angle, the area of that triangle is $<20$ as we have $\frac{1}{2} \cdot 40 \cdot \sin{\alpha}$ and $\sin$ is at most $1$. Note that for the other case, the side lengths around the obtuse angle must be $4$ and $x$ where we have $16+x^2 < 100 \rightarrow x < 2\sqrt{21}$. Using the same logic as the other case, the area is at most $4\sqrt{21}$. Square and add $4\sqrt{21}$ and $20$ to get the right answer \[a^2+b^2= \boxed{736}\Box\]

See also

2021 AIME II (Problems • Answer Key • Resources)
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Problem 6
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