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

Revision as of 18:39, 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

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 $8$ cm ( $AB$ ). Then, using a compass, draw 2 circles centered around points $A$ and $B$ , $7.5$ and $4.5$ 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).

$[asy] unitsize(0.25cm); pair A=(0,0), B=(8,0); path cA=circle(A,7.5), cB=circle(B,4.5); pair C=intersectionpoint(cA,cB); draw(A--B); draw(cA); draw(cB); draw(A--C--B--cycle); dot(A); dot(B); dot(C); label("A",A,S, fontsize(8)); label("B",B,S, fontsize(8)); label("C",C,N, fontsize(8)); label("8 cm",(A+B)/2, S, fontsize(6)); label("7.5 cm",(A+C)/2 + (-0.65,0.8),fontsize(6)); label("4.5 cm",(B+C)/2 + (1.55,-0.1),fontsize(6)); [/asy]$

After this, dropping the altitude to AB is simple with a ruler and careful placement, but the angle bisector can be a bit more complicated (Note: just like $AC$ an $BC$ , can be estimated with trial and error).

$[asy] unitsize(0.25cm); pair A=(0,0), B=(8,0); pair C=intersectionpoint(circle(A,7.5),circle(B,4.5)); draw(A--B--C--cycle); pair F=foot(C,A,B); draw(C--F); draw(rightanglemark(A,F,C,2)); pair U = unit(unit(A-B)+unit(C-B)); pair Q = B + 2.1*U; draw(B--Q, dashed); label("2.1 cm",(B+Q)/2,SE,fontsize(7)); dot(A); dot(B); dot(C); dot(F); label("A",A,S); label("B",B,S); label("C",C,N); label("F",F,S); [/asy]$

~shreyan.chethan (notes by curryswish)

Video Solution 1 by OmegaLearn

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

@@ Line 10: / Line 10: @@
 Let <imath>CD \perp AB</imath> with foot <imath>D</imath>. Right triangles <imath>ACD</imath> and <imath>BCD</imath> give <imath>AC^2 = AD^2+CD^2</imath>, <imath>BC^2 = BD^2+CD^2</imath>, <imath>AC^2-BC^2 = AD^2-BD^2 =(AD-BD)(AD+BD)</imath>.
-Since <imath>AD+BD = AB = 80</imath> and <imath>AC^2-BC^2 = 75^2-45^2 = 3600</imath>, substituting we get <imath>AD-BD = \frac{3600}{80} = 45</imath>, <imath>AD = \frac{125}{2}</imath>,
+Since <imath>AD+BD = AB = 80</imath> and <imath>AC^2-BC^2 = 75^2-45^2 = 3600</imath>, we get the equation <imath>3600 = 80(AD-BD)</imath>. This equation simplifies to <imath>45 = AD - BD</imath>. We can solve the system of equations <imath>AD + BD = 80</imath> and <imath>AD - BD = 45</imath> easily via elimination, and we get <imath>AD = \frac{125}{2}</imath>, <imath>BD = \frac{35}{2}</imath>. <imath>CD^2 = AC^2-AD^2 = 75^2-\left(\frac{125}{2}\right)^2 = \frac{6875}{4}</imath>, <imath>CD = \frac{25\sqrt{11}}{2}</imath>.
-<imath>BD = \frac{35}{2}</imath>. <imath>CD^2 = AC^2-AD^2 = 75^2-\left(\frac{125}{2}\right)^2 = \frac{6875}{4}</imath>, <imath>CD = \frac{25\sqrt{11}}{2}</imath>.
 By Angle Bisector Theorem, <imath>\frac{DP}{PC} = \frac{DB}{BC} = \frac{\frac{35}{2}}{45} = \frac{7}{18}</imath>, <imath>PC = CD-DP</imath> thus, <imath>18DP = 7(CD-DP)</imath>, <imath>25DP = 7CD</imath>, <imath>DP = \left(\frac{7}{25}\right)CD = \left(\frac{7}{25}\right)\left(\frac{25\sqrt{11}}{2}\right) = \frac{7\sqrt{11}}{2}</imath>. <imath>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</imath>, thus <imath>BP = \boxed{\text{(D) }21}.</imath>

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