All Questions ⁺⁰237784 Questions

0

40

1

+1540

Geometry

In triangle $ABC,$ $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.

learnmgcat Dec 12, 2024

0

46

1

+1540

Geometry

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?

learnmgcat Dec 12, 2024

0

37

1

+1540

Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

learnmgcat Dec 12, 2024

Dec 11, 2024

0

30

1

+898

Geometry

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

cooIcooIcooI17 Dec 11, 2024

0

38

1

+898

Geometry

In rectangle $EFGH$, let $M$ be the midpoint of $\overline{EF}$, and let $X$ be a point such that $MH = MX$, as shown below. If $\angle EFH = 30^\circ$ and $\angle MHX = 60^\circ,$ then find $\angle XHG,$ in degrees.

cooIcooIcooI17 Dec 11, 2024

0

21

1

+898

Geometry

Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1

cooIcooIcooI17 Dec 11, 2024

Dec 10, 2024

0

38

1

+613

Geometry

Triangle ABC has altitudes AD, BE, and CF. If AD=12, BE=16, and CF is a positive integer, then find the largest possible value of CF

Pythagorearn Dec 10, 2024

0

32

1

+613

Geometry

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.

Pythagorearn Dec 10, 2024

0

33

1

+774

Algebra

Suppose a does not equal 0. Compute log base a (ab^2) in terms of a and b.

siIviajendeukie Dec 10, 2024

0

36

1

+498

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

MeIdHunter Dec 10, 2024

0

33

1

+498

Geometry

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

MeIdHunter Dec 10, 2024

0

47

0

+498

Geometry

In parallelogram EFGH, let M be the point on \overline{EF} such that FM:ME = 1:1, and let N be the point on \overline{EH} such that HN:NE = 1:1. Line segments \overline{FH} and \overline{GM} intersect at P, and line segments \overline{FH} and GN intersect read more ..

MeIdHunter Dec 10, 2024

0

48

0

+498

Geometry

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (2,4),$ $B = (-3,1),$ and $C = (1,7)$, then find the constant read more ..

MeIdHunter Dec 10, 2024

0

33

2

+9

Logarithms

Suppose a does not equal 0. Compute log base 8a (4^b) in terms of a and b.

hemii18675 Dec 10, 2024

0

40

1

+823

Geometry

A rectangle in the coordinate plane is shown below. A line with slope 2 splits this rectangle into two regions with equal area. What is the y-intercept of this line?
The rectangle has coordinates (-1,-1), (-1,1), (1,1), read more ..

crimefightingvigiI Dec 10, 2024

0

51

0

+823

Geometry

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $11,$ and the slope of line segment $\overline{OQ}$ is $13.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

crimefightingvigiI Dec 10, 2024

0

38

1

+823

Geometry

A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The circle is x^2 + y^2 = 1, and the line is y = x.

crimefightingvigiI Dec 10, 2024

0

48

0

+823

Geometry

Chords UV, WX, and YZ of a circle are parallel. The distance between chords UV and WX is 1, and the distance between chords WX and YZ is also 1. If UV =6 and YZ = 4, then find WX.

crimefightingvigiI Dec 10, 2024

0

44

0

+1540

Geometry

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

learnmgcat Dec 10, 2024

0

45

0

+1540

Geometry

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

learnmgcat Dec 10, 2024

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