Practice (41)

2460

Show that the triangles

$ABC$ ,

$ABH$ ,

$BCH$ , and

$CAH$ all have the same nine-point circle, providing that the orthocenter

$H$ does not coincide with each of the vertices

$A$ ,

$B$ ,

$C$ .

2462

Let

$a, b, c$ be respectively the lengths of three sides of a triangle, and

$R$ be the triangle's circumradius. Show that

$R = \frac{abc}{\sqrt{(a+b+c)(b+c-a)(c+a-b)(b+a-c)}}$

2463

Prove the angle bisector theorem

2465

Prove the median's length formula:

$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$

2466

Prove the triangle's altitude formula:

$h_a = \frac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}$

2467

Prove the triangle's angle bisector length formula:

$t_a=\frac{2}{b+c}\sqrt{bcp(p-a)}=\sqrt{bc\Big(1-(\frac{a}{b+c})\Big)^2}$

2499

Triangle

$ABC$ is isosceles, and

$\angle{ABC}=x^\circ$ . If the sum of possible measurements of

$\angle{BAC}=240^\circ$ , find

$x^\circ$ .

2502

Let

$ABCD$ be a quadrilateral with an inscribed circle

$\omega$ that has center

$I$ . If

$IA=5$ ,

$IB=7$ ,

$IC=4$ ,

$ID=9$ , find the value of

$\frac{AB}{CD}$ .

2556

There are

$n$ circles inside a square

$ABC$ whose side's length is

$a$ . If the area of any circle is no more than 1, and every line that is parallel to one side of

$ABCD$ intersects at most one such circle, show that the sum of the area of all these

$n$ circles is less than

$a$ .

2559

Point $O$ is the center of the regular octagon $ABCDEFGH$ , and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?

2565

What is the smallest whole number larger than the perimeter of any triangle with a side of length

$5$ and a side of length

$19$ ?

2629

(Stewart's Theorem) Show that $b^2m + c^2n = a(d^2 +mn)$

2630

What is the angle bisector's theorem?

2649

How many different triangles have vertices selected from the seven points (-4, 0), (-2, 0), (0,0), (2,0), (4,0), (0,2), and (0,4)?

2650

Three circular cylinders are strapped together as shown. The cross-section of each cylinder is a circle of radius 1. Presuming that the strap used to bind the cylinders together has no thickness and no extra length, how long is the binding strap?

2674

Let

$\triangle{ABC}$ be a Pythagorean triangle. If

$\triangle{ABC}$ 's circumstance is 30, find its circumcircle's area.

2768

In triangle

$ABC$ , side

$AC$ and the perpendicular bisector of

$BC$ meet in point

$D$ , and

$BD$ bisects

$\angle ABC$ . If

$AD=9$ and

$DC=7$ , what is the area of triangle ABD?

2772

Find the degree measure of an angle whose complement is 25% of its supplement.

2775

$45^\circ$ arc of circle A is equal in length to a

$30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?

2786

Let

$C_1$ and

$C_2$ be circles defined by

$(x-10)^2 + y^2 = 36$ and

$(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment

$PQ$ that is tangent to

$C_1$ at

$P$ and to

$C_2$ at

$Q$ ?

2813

In the diagram , $PA = QB = PC = QC = PD = QD = 1, CE = CF = EF$ and $EA = BF = 2AB$ . Determine $BD$ .

2825

In rectangle $ABCD$ , we have $AB=8$ , $BC=9$ , $H$ is on $BC$ with $BH=6$ , $E$ is on $AD$ with $DE=4$ , line $EC$ intersects line $AH$ at $G$ , and $F$ is on line $AD$ with $GF \perp AF$ . Find the length of $GF$ .

2834

A line that passes through the origin intersects both the line

$x=1$ and the line

$y=1+\frac{\sqrt{3}}{3}x$ . The three lines create an equilateral triangle. What is the perimeter of the triangle?

2858

Let

$\triangle{ABC}$ be a right triangle whose sides lengths are all integers. If

$\triangle{ABC}$ 's perimeter is 30, find its incircle's area.

2883

In rectangle

$ABCD,$

$AB=6$ and

$BC=3$ . Point

$E$ between

$B$ and

$C$ , and point

$F$ between

$E$ and

$C$ are such that

$BE=EF=FC$ . Segments

$\overline{AE}$ and

$\overline{AF}$ intersect

$\overline{BD}$ at

$P$ and

$Q$ , respectively. The ratio

$BP:PQ:QD$ can be written as

$r:s:t$ where the greatest common factor of

$r,s$ and

$t$ is 1. What is

$r+s+t$ ?

PlaneGeometry Triangle PythagoreanTheorem AngleBisectorTheorem StewartTheorem TriangleCenter SimiarTriangle Circle AreaMethod CoordinatedGeometry

AMC8 AMC10/12

With Solutions

back to index | new