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2025 AMC 12A Problems/Problem 16: Difference between revisions

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After all of this, we can reuse our ruler and measure <imath>BP = 2.1\,\text{cm}</imath>, and remembering our old scale of <imath>80=8\,\text{cm}</imath>, our final answer is <imath>\boxed{\text{(D) }21}.</imath>
After all of this, we can reuse our ruler and measure <imath>BP = 2.1\,\text{cm}</imath>, and using our old scale of <imath>80=8\,\text{cm}</imath>, our final answer is <imath>\boxed{\text{(D) }21}.</imath>




Revision as of 18:50, 8 November 2025

The following problem is from both the 2025 AMC 10A #23 and 2025 AMC 12A #16, so both problems redirect to this page.

Problem

Triangle $\triangle ABC$ has side lengths $AB = 80$, $BC = 45$, and $AC = 75$. The bisector $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P$. What is $BP$?

$\textbf{(A)}~18\qquad\textbf{(B)}~19\qquad\textbf{(C)}~20\qquad\textbf{(D)}~21\qquad\textbf{(E)}~22$

Solution 1

Let $CD \perp AB$ with foot $D$. Right triangles $ACD$ and $BCD$ give $AC^2 = AD^2+CD^2$, $BC^2 = BD^2+CD^2$, $AC^2-BC^2 = AD^2-BD^2 =(AD-BD)(AD+BD)$.

Since $AD+BD = AB = 80$ and $AC^2-BC^2 = 75^2-45^2 = 3600$, we get the equation $3600 = 80(AD-BD)$. This equation simplifies to $45 = AD - BD$. We can solve the system of equations $AD + BD = 80$ and $AD - BD = 45$ easily via elimination, and we get $AD = \frac{125}{2}$, $BD = \frac{35}{2}$. $CD^2 = AC^2-AD^2 = 75^2-\left(\frac{125}{2}\right)^2 = \frac{6875}{4}$, $CD = \frac{25\sqrt{11}}{2}$.

By Angle Bisector Theorem, $\frac{DP}{PC} = \frac{DB}{BC} = \frac{\frac{35}{2}}{45} = \frac{7}{18}$, $PC = CD-DP$ thus, $18DP = 7(CD-DP)$, $25DP = 7CD$, $DP = \left(\frac{7}{25}\right)CD = \left(\frac{7}{25}\right)\left(\frac{25\sqrt{11}}{2}\right) = \frac{7\sqrt{11}}{2}$. $BP^2 = BD^2+DP^2 = \left(\frac{35}{2}\right)^2+\left(\frac{7\sqrt{11}}{2}\right)^2 = \frac{1225}{4}+\frac{49(11)}{4} = \frac{1764}{4} = 441$, thus $BP = \boxed{\text{(D) }21}.$

~pigwash ~aldzandrtc(fixed logical jump)

Solution 2 (Law of Cosines)

Scale this down to a $9-15-16$ triangle (we will multiply the result by $5$ in the end). Note that $9^2+16^2-18*16*\cos(\angle B) = 15^2$, so $112 = 18 * 16 * \cos(\angle B)$, which simplifies to $\frac{7}{18} = \cos(\angle B)$. Then $\cos(\frac{\angle B}{2}) = \sqrt{\frac{1+\frac{7}{18}}{2}} = \sqrt{\frac{25}{36}} = \frac{5}{6}$ (positive root since the angle is acute). Therefore, we have $\frac{BD}{BP} = \frac{5}{6}$, assuming that $D$ is the foot of the altitude. There are many ways to proceed from here to find $BD$. Note that by Heron's formula, the area of the scaled-down triangle is $\sqrt{(20)(4)(5)(11)} = 20\sqrt{11}$. Therefore, $CD = \frac{5}{2}\sqrt{11}$. Using Pythagorean Theorem, we get $BD= \sqrt{81-\frac{25 \cdot 11}{4}} = \sqrt{\frac{324-275}{4}} = \frac{7}{2}$. Therefore, we get $\frac{\frac{7}{2}}{BP} = \frac{5}{6}$, so $BP = \frac{21}{5}$, and we scale up by $5$ to get $\boxed{21}$. ~ScoutViolet

Solution 3 (Stewarts)

Let the foot of the altitude coming from $C$ on segment $AB$ be $D$. Using the fact that $CD$ is a common leg in right triangles $\triangle CDB$ and $\triangle CDA$, we have \[45^2 - BD^2 = 75^2 - (80-BD)^2.\] Expanding gives \[45^2=75^2-80^2+160BD,\] so $BD=\frac{35}{2}.$ Let the foot of the angle bisector from $B$ to $AC$ be point $E$. Since $BE$ is the angle bisector of $\angle CBA$, we can use the angle bisector theorem. This gives \[\frac{45}{CE}=\frac{80}{75-CE},\] so $CE=27$ and $AE=75-27=48$. Now we can use Stewart’s Theorem to find $\overline{BE}$. We have \[(48)(45)^2+(27)(80)^2=(27)(48)(75)+75BE^2.\] To simplify this expression, just divide by the greatest common divisor and solve from there. In the end, we get $BE=48$. Let $BP=x$, so $PE=48-x$. Draw the altitude from $E$ down to $AB$. Let the foot of this altitude be $F$. Since $EF || CD$, we have $\triangle AFE \sim \triangle ADC$. Hence, we can write the equation \[\frac{48}{75}=\frac{AF}{\frac{125}{2}}.\] Solving gives $AF=40$, so $FD=\frac{125}{2}-40=\frac{45}{2}$. Since $PD || EF$, we also have $\triangle BDP \sim \triangle BFE$, so we have \[\frac{x}{48}=\frac{\frac{35}{2}}{40}.\] Solving for $x$ gives $\boxed{x=21}$ or $\boxed{\text{D}}.$

~evanhliu2009

Solution 4 (Rulerbash)

Preface: When we have a problem as such, involving a simple diagram with minimal instructions, I use a method I named "rulerbash". Rulerbash should only be used in specific cases and as a last resort, mainly in the event of a time crunch or a difficult problem.

Start with the longest side, drawing a line with a length of $8\,\text{cm}$ ($AB$). Then, using a compass, draw 2 circles centered around points $A$ and $B$, $7.5\,\text{cm}$ and $4.5\,\text{cm}$ radiuses respectfully. At the point of intersection of these 2 circles, we have point $C$, completing a perfectly scaled drawing of $\triangle ABC$. (Note the circles are not necessary with a bit of trial and error with the side lengths, they simply offer a way to get it done first try).

[asy] unitsize(0.25cm); pair A=(0,0), B=(8,0); path cA=circle(A,7.5), cB=circle(B,4.5); pair C=intersectionpoint(cA,cB); draw(A--B); draw(cA); draw(cB); draw(A--C--B--cycle); dot(A); dot(B); dot(C); label("A",A,S, fontsize(8)); label("B",B,S, fontsize(8)); label("C",C,N, fontsize(8)); label("8 cm",(A+B)/2, S, fontsize(6)); label("7.5 cm",(A+C)/2 + (-0.85,0.8),fontsize(6)); label("4.5 cm",(B+C)/2 + (1.75,-0.1),fontsize(6)); [/asy]



After this, dropping the altitude to $AB$ is simple with a ruler and careful placement, and angle bisector can be estimated quite accurately.

[asy] unitsize(0.55cm); import olympiad; pair A=(0,0), B=(8,0); pair C=intersectionpoint(circle(A,7.5),circle(B,4.5)); draw(A--B--C--cycle); pair F=foot(C,A,B); draw(C--F); draw(rightanglemark(A,F,C,2)); pair U=unit(unit(A-B)+unit(C-B)); pair P=intersectionpoint(B--(B+100*U), C--F); draw(B--P,dashed); dot(A);dot(B);dot(C);dot(P); label("A",A,S,fontsize(12)); label("B",B,S,fontsize(12)); label("C",C,N,fontsize(12)); label("P",P,NE,fontsize(12)); [/asy]

After all of this, we can reuse our ruler and measure $BP = 2.1\,\text{cm}$, and using our old scale of $80=8\,\text{cm}$, our final answer is $\boxed{\text{(D) }21}.$


~shreyan.chethan (notes by curryswish)

Video Solution 1 by OmegaLearn

https://youtu.be/DI-q_-cYMVU

See Also

2025 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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
2025 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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

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