Practice (59)

3022

Let point $P$ be inside equilateral $\triangle{ABC}$ such that $\angle{APB}=115^\circ$ and $\angle{BPC}=125^\circ$. Find the measurements of the three internal angles if we construct a triangle using $PA$, $PB$, and $PC$.

3023

Let $\triangle{ABC}$ be an isosceles right triangle where $\angle{C}=90^\circ$. If points $M$ and $N$ are on $AB$ such that $\angle{MCN}=45^\circ$, $AM=4$, and $BN=3$, find the length of $MN$.

3024

Given an acute $\triangle$, let $D$ be the middle point of $AB$, and $DE\perp DF$ where points $E$ and $F$ are on the other two sides respectively. Show that $S_{\triangle{DEF}} < S_{\triangle{ADF}} + S_{\triangle{BDE}}$

3036

Fagnoan's Problem

3037

Fermat's Point

3038

Let $P$ be a point inside parallelogram $ABCD$. If $\angle{PAB}=\angle{PCB}$, show $\angle{PBA} = \angle{PDA}$.

3096

As shown, $D$ is the midpoint of $BC$. Point $E$ is on $AD$ such that $BE=AF$. Show that $AF=EF$.

3103

Let $ABCDE$ be a pentagon such that $AB=BC=CD=DE=EA$ as shown. If $\angle{ABC}=2\angle{DBE}$, find the measurement of $\angle{ABC}$.

3144

In tetrahedron $ABCD$, $\angle{ADB} = \angle{BDC} = \angle{CDA} = 60^\circ$, $AD=BD=3$, and $CD=2$. Find the radius of $ABCD$'s circumsphere.

3309

Let point $P$ inside an equilateral $\triangle{ABC}$ such that $AP=3$, $BP=4$, and $CP=5$. Find the side length of $\triangle{ABC}$.

3936

As shown, both $ABCD$ and $OPRQ$ are squares. Additionally, $O$ is the center of $ABCD$, $OP=1$, $BP=\sqrt{2}$, and $CQ=\sqrt{5}$. Find the length of $DR$.

4534

Consider the ellipse $x^2+\frac{y^2}{4}=1$. What is the area of the smallest diamond shape with two vertices on the $x$-axis and two vertices on the $y$-axis that contains this ellipse?

4749

Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$

PlaneGeometry PythagoreanTheorem StewartTheorem RotationMethod SolidGemoetry LawOfCosines

AMC10/12 UConn

With Solutions

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