Geometry

In the diagram, line $L_1$ is parallel to line $L_2$. Find the value of $x$.

Find angle $p$ in the diagram showing vertically opposite angles.

Determine the value of angle $x$ on the straight line $L$.

Find the size of angle $b$ in the quadrilateral.

Lines $AB$ and $CD$ are parallel. Transversal $EF$ intersects them at $G$ and $H$. If $\angle AGH = 2x + 10$ and $\angle GHD = 3x - 20$, find $x$.

In triangle $XYZ$, the ratio of angles $\angle X : \angle Y : \angle Z$ is $1:2:3$. Find the largest angle.

If the exterior angle of a regular polygon is $24^{\circ}$, how many sides does it have?

In a star-shaped badge (a pentagram), the five points are vertices of triangles attached to a central regular pentagon. If the pentagon in the center is regular, what is the sum of the five angles at the tips of the star?

An irregular hexagon has five interior angles that are all equal to $110^{\circ}$. Calculate the size of the sixth interior angle.

A technician is aligning two laser beams that must be parallel. Beam A and Beam B are crossed by a third calibration beam. The technician measures two consecutive interior angles on the same side of the calibration beam as $4y + 12^{\circ}$ and $6y - 2^{\circ}$. For the beams to be perfectly parallel, what must be the value of $y$?

A shape $P$ is transformed to shape $Q$ as shown in the coordinate plane. Which vector represents the translation that moves shape $P$ to shape $Q$?

What translation vector moves a point from $(0,0)$ to $(-3, 5)$?

If shape X is rotated 90° anticlockwise about the origin, which quadrant will the image be in?

Which transformation is shown where the orientation of the shape is reversed?

Shape $A$ is mapped to $B$ by a rotation of 90° anticlockwise about the origin. Shape $B$ is mapped to $C$ by a reflection in the x-axis. If $A$ has vertex $(1, 0)$, what is the corresponding vertex in $C$?

The point $(-2, 4)$ is reflected in the line $y = 1$. What is the image point?

How many lines of symmetry does the shape formed by the function $y = x^2$ have (restricted to $-2 \le x \le 2$)?

Point $P(x, y)$ is rotated $90^{\circ}$ anti-clockwise about the origin to $P'(-3, 5)$. What were the original coordinates of point $P$?

A rectangle with area $10 \text{ cm}^2$ is enlarged. The new area is $160 \text{ cm}^2$. If the center of enlargement is the origin, what is the scale factor $k$?

Triangle $LMN$ has vertices $L(1, 1)$, $M(1, 4)$, and $N(3, 1)$. If it is reflected in the line $x = 1$, which statement correctly describes the resulting image?

A right-angled triangle has two shorter sides of lengths $6$ cm and $8$ cm. Calculate the length of the hypotenuse, denoted by $x$.

In a right-angled triangle, the legs are $9$ m and $12$ m. Find the hypotenuse.

A square has a diagonal of length $\sqrt{50}$ cm. What is the length of one side of the square?

Find the length of the side labeled $y$ in the triangle below.

In the diagram, $BCD$ is a straight line. Triangle $ABC$ is right-angled at $B$. $AB = 9$ cm, $AC = 15$ cm, and $AD = 41$ cm. Find the length of $CD$.

A rhombus has diagonals of length $12$ cm and $16$ cm. What is the length of one side of the rhombus?

Calculate the area of a right-angled triangle with a hypotenuse of $10$ cm and one leg of $6$ cm.

A television screen size is given by the length of its diagonal. If a TV screen is $48$ inches wide and $36$ inches high, what is the advertised size of the TV screen in inches?

A gardener wants to create a path diagonally across a rectangular garden measuring $40$ m by $30$ m. If the path costs $15$ per meter to build, what is the total cost of the diagonal path?

Find the value of $x$ in the diagram where two right-angled triangles share a common side. The first triangle has legs $3$ and $4$. The second triangle has the hypotenuse of the first as one leg and $12$ as the other leg. $x$ is the hypotenuse of the second triangle.

If a circle has an area of $49\pi$ cm$^2$, what is its circumference?

In the diagram, if the radius is $x$, which expression represents the area?

What is the relationship between the radius ($r$) and the diameter ($d$) of a circle?

Calculate the area of a circle where the diameter is $6$ cm. Leave your answer in terms of $\pi$.

What is the area of a circle that has the same numerical value for its circumference and its area?

Two circles are placed side by side so they touch at one point. If both circles have a radius of $5$ cm, what is the length of the rectangle that just encloses them?

Calculate the area of the shaded region between the square and the inscribed circle. The square has a side of $14$ cm. Use $\pi = \frac{22}{7}$.

A landscape architect is designing a circular flower bed surrounded by a gravel path. The inner flower bed has a radius of $6$ meters. The gravel path has a constant width of $w$ meters. If the area of the gravel path is exactly twice the area of the inner flower bed, calculate the total radius of the entire structure (flower bed plus path). Give your answer to two decimal places.

The perimeter of a shaded region formed by a large semicircle of diameter $20\text{ cm}$ and two smaller identical semicircles of diameter $10\text{ cm}$ placed along the large diameter is required. Calculate the total perimeter of this shaded boundary.

A circular piece of card has a radius of $10\text{ cm}$. A concentric circle of radius $x$ is cut out from the center. If the area of the remaining annulus is $64\pi \text{ cm}^2$, find the value of $x$.

In the construction of a perpendicular from a point $P$ to a line $L$, where $P$ is not on $L$, what is the first step?

Which tool is essential for drawing a locus of points at a constant distance from a fixed point?

What geometric construction is used to find the center of a circle if you are given three points on its circumference?

If you bisect a $90^{\circ}$ angle, and then bisect one of the resulting angles, what is the size of the smallest angle created?

In the diagram, a straight line segment $AB$ is $8\text{ cm}$ long. A point $P$ moves such that it is always equidistant from point $A$ and point $B$. Which geometrical construction represents the locus of point $P$?

The locus of points equidistant from two parallel lines $L_1$ and $L_2$ that are $6\text{ units}$ apart is:

How many points are both $5\text{ cm}$ from point $A$ and $5\text{ cm}$ from point $B$, if $AB = 10\text{ cm}$?

In a square $ABCD$ of side length $10$, a locus is formed by all points $P$ such that the distance from $P$ to $AD$ is equal to the distance from $P$ to $AB$. Which geometric feature represents this locus?

Given a fixed segment $AB$, a point $C$ moves such that the area of triangle $ABC$ is always $20\text{ cm}^2$. If $AB = 8\text{ cm}$, what is the locus of $C$?

A point $X$ moves such that it is always $3\text{ cm}$ from a line segment $AB$ which is $10\text{ cm}$ long. What is the shape of the locus of $X$?