Geometry

Two lines intersect such that one angle is $x$ and its adjacent angle is $2x + 30^\circ$. Find $x$.

Find the sum of the exterior angles of a decagon.

In the diagram, $m \parallel n$. Find the value of $y$.

If the exterior angle of a triangle is $130^\circ$ and one of the opposite interior angles is $50^\circ$, find the other opposite interior angle.

The exterior angles of a triangle are in the ratio $2:3:4$. Find the smallest interior angle of the triangle.

In the diagram, $AB \parallel CD$. The angle $\angle BAE = 40^\circ$ and $\angle DCE = 30^\circ$. Find the reflex angle $\angle AEC$.

In a hexagon, four of the interior angles are $120^\circ$ each. The other two angles are equal. Find the size of one of the equal angles.

In a tiling pattern, regular octagons and regular $k$-gons meet at a vertex such that two octagons and one $k$-gon perfectly surround the point. Determine the value of $k$.

The exterior angles of a polygon are $x, 2x, 3x, 4x, 5x, 6x, 7x, 8x$. Find the value of the smallest interior angle.

A roof truss is being built where the top chord $AC$ is parallel to the bottom chord $DF$. Vertical struts $BE$ and $CF$ are installed. If $\angle ABE = 118^{\circ}$ and $\angle BCF = 105^{\circ}$, what is the measure of the angle $\angle EBC$ if $BE$ is perpendicular to $AC$?

The figure shows a Kite. How many lines of symmetry does it have?

Which coordinate axis is the line of symmetry for the curve $y = \cos(x)$ shown in the interval $[-\pi, \pi]$?

In the diagram, a shape is formed by two overlapping congruent circles. What is the order of rotational symmetry of the shaded region?

A square is rotated about its center. At what minimum angle (greater than $0^{\circ}$) will the square first map onto itself?

The diagram shows a rectangle with two identical semi-circles removed from opposite sides. What is the order of rotational symmetry of the remaining figure?

A curve has the equation $y = |x|$. Which line is a line of symmetry for this graph?

The diagram shows a 'star' polygon (a hexagram). What is its order of rotational symmetry?

A piece of sheet metal is cut into a shape bounded by $y = \sqrt{x}$ and $y = x^3$ for $0 \le x \le 1$. The shape is then reflected across the line $y = x$ to double the area. What is the total area of the resulting symmetric shape?

An elliptical track is given by the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. A camera is placed at one of the foci. How many lines of symmetry does the elliptical track itself possess?

A wave profile is described by $y = 2\sin(2x - \frac{\pi}{2})$. A student claims the graph is symmetric about the y-axis. Determine if this is true and identify the property.

The ratio of the corresponding altitudes of two similar triangles is $3:4$. If the area of the larger triangle is $80$, what is the area of the smaller triangle?

Two spheres have surface areas in the ratio $9:16$. Find the ratio of their radii.

In the diagram, $AB || CD$. If $OA=3$, $OC=6$, and $OB=2$, find $OD$.

Two similar cylinders have heights $h_1 = 5 \text{ cm}$ and $h_2 = 10 \text{ cm}$. If the lateral surface area of the smaller cylinder is $50\pi \text{ cm}^2$, what is the lateral surface area of the larger one?

In the diagram below, $ABC$ is a triangle where $DE$ is parallel to $BC$. Given that $AD = 6 \text{ cm}$, $DB = 4 \text{ cm}$, and the area of triangle $ADE$ is $54 \text{ cm}^2$, calculate the area of the quadrilateral $DBCE$.

Triangle $ABC$ has side lengths $5, 12, 13$. Triangle $PQR$ is similar to $ABC$ and has a perimeter of $90$. What is the length of the shortest side of triangle $PQR$?

In the diagram, $\triangle ADE \sim \triangle ABC$. If $DE = 4$, $BC = 10$, and the area of $\triangle ADE$ is $16$, find the area of the trapezoid $DBCE$.

Two cylinders are mathematically similar. The smaller cylinder has a height of $10\text{ cm}$ and a lateral surface area of $60\pi\text{ cm}^2$. The larger cylinder has a lateral surface area of $135\pi\text{ cm}^2$. Determine the volume of the larger cylinder if the smaller one has a volume of $90\pi\text{ cm}^3$.

A map has a scale of $1:25,000$. A rectangular forest is represented on the map by a rectangle of area $16\text{ cm}^2$. Calculate the actual area of the forest in square kilometers ($1\text{ km}^2 = 10^{10}\text{ cm}^2$ is incorrect, remember $1\text{ km} = 10^5\text{ cm}$).

In a coordinate plane, $\triangle OPQ$ has vertices $O(0,0)$, $P(6,0)$, and $Q(0,8)$. A second triangle $OR S$ is similar to $OPQ$ with a scale factor of $k = 1.5$ centered at the origin. What is the length of the segment $RS$?

In the diagram, points $A$, $B$, and $C$ lie on the circumference of a circle with center $O$. Given that the angle $\angle ABC = 54^{\circ}$, what is the size of the angle $\angle AOC$ subtended by the same arc $AC$ at the center?

In a circle, chords $PQ$ and $RS$ are equal in length. If the distance of $PQ$ from the center $O$ is $5$ cm, what is the distance of $RS$ from the center?

A circle is inscribed in a square of side length $10$ cm. What is the area of the circle in terms of $\pi$?

Chords $AB$ and $CD$ of a circle intersect at point $E$ inside the circle. If $AE = 4$ cm, $EB = 9$ cm, and $CE = 6$ cm, find the length of $ED$.

Two parallel chords of a circle are $6$ cm and $8$ cm long. If they are on the same side of the center and the radius of the circle is $5$ cm, find the distance between the chords.

A circle passes through the vertices of an equilateral triangle of side $6$ cm. Find the radius of the circle.

In the diagram, $O$ is the center of the circle and $PT$ is a tangent at $T$. If $\angle TPO = 30^\circ$ and $OT = 5$ cm, find the length of $OP$.

A surveyor identifies four points $J, K, L, M$ on a circular boundary. The diagonals $JL$ and $KM$ are diameters of the circle. What type of quadrilateral is $JKLM$?

A right-angled triangle $ABC$ has $\angle ABC = 90^{\circ}$. A circle is drawn with $AC$ as the diameter. Which of the following is true regarding point $B$?

Two circles intersect at points $P$ and $Q$. A straight line through $P$ intersects the first circle at $A$ and the second circle at $B$. A straight line through $Q$ intersects the first circle at $C$ and the second circle at $D$. If $\angle PAC = 70^{\circ}$, calculate $\angle QDB$.

In a circle of radius 5, a point $M$ moves such that it is always 2 units from the circumference (internally). What is the locus of $M$?

A point $P$ moves such that the distance $PA$ is always equal to $3$, where $A$ is the fixed point $(1, 1)$. What is the equation of the locus?

In a coordinate system, what is the locus of a point that is equidistant from the x-axis and the y-axis?

Which diagram correctly illustrates the locus of points $z$ satisfying $arg(z - 1) = \frac{\pi}{4}$?

A point $P$ moves in the Cartesian plane such that its distance from a fixed point $A(2, 2)$ is always equal to its distance from the line $y = -2$. The diagram shows the point $A$ and the directrix line $L$. Which of the following equations describes the locus of $P$?

Find the equation of the locus of a point $P(x, y)$ such that its distance from the y-axis is equal to its distance from the point $(4, 0)$.

The locus of points $(x, y)$ such that $|x| + |y| = 4$ forms a geometric shape. What is this shape?

A mechanical linkage consists of a rod $OP$ of length 10 cm, where $O$ is the fixed origin $(0,0)$. Another rod $PQ$ of length 6 cm is attached to $P$. If $P$ moves along the circle $x^2 + y^2 = 100$ and $Q$ is always constrained to lie on the $x$-axis, find the maximum possible $x$-coordinate of $Q$.

A point $P(x,y)$ moves such that its distance from the line $x = -2$ is always equal to its distance from the point $F(2,0)$. Find the y-coordinate of the point(s) on this locus when $x = 6$.

Consider a square field with vertices at $(0,0)$, $(10,0)$, $(10,10)$, and $(0,10)$. A goat is tethered at the midpoint of the side $(0,0)$ to $(10,0)$ with a rope of length 7. Another region is defined by points closer to the corner $(10,10)$ than to the corner $(0,10)$. What is the shape of the boundary of the region where the goat can graze that is also closer to $(10,10)$ than $(0,10)$?