Geometry and measures

In triangle $PQR$, which vertex is opposite the side $PR$?

Lines $AB$ and $CD$ are parallel. A line through $E$ meets $AB$ at $F$ at a right angle. State the relationship between $EF$ and $CD$.

Draw quadrilateral $ABCD$ with $AB\parallel CD$. Draw diagonal $AC$. Mark a point $E$ on $AC$, then draw the line through $E$ perpendicular to $AB$, meeting $AB$ at $F$ and $CD$ at $G$.

Describe how to construct the perpendicular bisector of a line segment $AB$ using a ruler and compasses.

Two straight paths meet at $O$. A lamp must be equally distant from the two paths and no more than $6$ m from $O$. Describe the locus of possible positions inside the angle between the paths.

Points $A$ and $B$ are $8$ cm apart. A point $P$ must satisfy $PA=PB$ and $PA\leq5$ cm. Describe and construct the complete locus of $P$.

Two adjacent angles on a straight line are $63^\circ$ and $x^\circ$. Work out $x$.

Two parallel lines are crossed by a transversal. A pair of alternate angles are $(4x+7)^\circ$ and $(7x-38)^\circ$. Find $x$ and the size of these angles.

A polygon has $13$ sides. Twelve of its interior angles are each $155^\circ$. Work out the remaining interior angle.

Name the quadrilateral that has exactly one pair of parallel sides.

One interior angle of a rhombus is $68^\circ$. Work out the other three interior angles.

The diagonals of quadrilateral $WXYZ$ bisect each other and are equal in length. Explain why $WXYZ$ must be a rectangle, and why the information does not prove that it is a square.

Triangle $ABC$ has side lengths $5$ cm, $7$ cm and $9$ cm. Triangle $PQR$ has side lengths $5$ cm, $7$ cm and $9$ cm. State the congruence criterion.

Triangles $ABC$ and $DEF$ are right-angled at $B$ and $E$. Also $AC=DF=13$ cm and $AB=DE=5$ cm. Explain why the triangles are congruent.

In quadrilateral $ABCD$, diagonal $AC$ is drawn. Given that $AB=AD$ and $\angle BAC=\angle CAD$, prove that $BC=CD$.

Triangle $ABC$ is isosceles with $AB=AC$. Angle $B$ is $47^\circ$. Work out angle $A$.

A right-angled triangle has perpendicular sides $9$ cm and $12$ cm. Work out the length of its hypotenuse.

In quadrilateral $ABCD$, $AB\parallel CD$ and $AB=CD$. Diagonal $AC$ is drawn. Prove that $BC=AD$.

Point $A(-3,4)$ is translated by the vector $\begin{pmatrix}5\\-2\end{pmatrix}$. Find the coordinates of its image.

Point $P(4,-1)$ is rotated $90^\circ$ anticlockwise about the origin. Find the coordinates of its image $P'$.

Triangle $ABC$ has vertices $A(6,3)$, $B(0,5)$ and $C(-2,-1)$. It is enlarged by scale factor $-\frac12$ about centre $(2,-1)$. Find the three image coordinates.

A shape is rotated and then translated. State two properties of the shape that must remain invariant.

A shape is translated first by $\begin{pmatrix}3\\-2\end{pmatrix}$ and then by $\begin{pmatrix}-5\\7\end{pmatrix}$. Describe the single equivalent transformation.

A shape is reflected in the $x$-axis and then reflected in the line $y=x$. Describe the single equivalent transformation and state whether orientation is preserved.

A circle has radius $6.5$ cm. Write down its diameter.

A straight segment joins two points on a circle but does not pass through its centre. Another straight line meets the circle at exactly one point. Give the geometric name of each.

A circle is centred at $O$. Distinct points $A$ and $B$ are on its circumference, and $AB$ is not a diameter. Describe precisely the boundaries of the minor sector $AOB$ and the minor segment cut off by $AB$.

$AB$ is a diameter of a circle and $C$ is another point on the circumference. Find $\angle ACB$.

In a circle with centre $O$, the angle $AOB$ is $124^\circ$. Point $C$ lies on the major arc $AB$. Work out $\angle ACB$.

From a point $P$ outside a circle with centre $O$, tangents touch the circle at $A$ and $B$. Prove that $PA=PB$ and that $OP$ bisects $\angle APB$.

Find the midpoint of the line segment joining $(-5,2)$ to $(7,8)$.

Points $A(-1,4)$ and $B(5,-4)$ are joined. Find the exact length $AB$.

Triangle $ABC$ has vertices $A(-2,1)$, $B(4,3)$ and $C(2,9)$. Show that the triangle is right-angled and isosceles, then work out its area.

Write down the number of faces and the number of vertices of a triangular prism.

A solid has one circular plane face, one curved surface, one circular edge and one vertex. Name the solid.

A prism has a regular $9$-sided polygon as each end. Work out its numbers of faces, edges and vertices.

A cuboid is $8$ cm long, $5$ cm wide and $3$ cm high. State the shape and dimensions of its plan.

A numbered plan of unit-cube stacks has two rows. From north to south, the rows are $(2,1)$ and $(3,0)$, with entries ordered west to east. Give the visible column heights in the front elevation viewed from the south and in the side elevation viewed from the east.

A solid uses vertical stacks of unit cubes on all four cells of a $2$ by $2$ plan. Its front elevation viewed from the south has heights $(3,2)$ from west to east. Its side elevation viewed from the east has heights $(2,3)$ from north to south. Find the least possible number of cubes and give one numbered plan that achieves it.

Convert $3.75\text{ kg}$ to grams.

A rectangular water tank is $0.8\text{ m}$ long, $0.5\text{ m}$ wide and $0.6\text{ m}$ high. It is $65\%$ full. Work out the volume of water in litres.

A floor measures $4.8\text{ m}$ by $3.6\text{ m}$. Square tiles have side length $30\text{ cm}$ and are sold in boxes of $12$. A decorator buys at least $8\%$ more tiles than the exact number needed. Each box costs £$14.75$. Work out the total cost.

A ray from point $P$ makes an angle of $68^\circ$ clockwise from north. Write its bearing from $P$.

On a map with north at the top, point $B$ is $6\text{ cm}$ east and $8\text{ cm}$ north of point $A$. The scale is $1:20\,000$. Work out the real straight-line distance from $A$ to $B$ and the bearing of $B$ from $A$. Give the bearing to the nearest degree.

On a scale drawing, a road of actual length $3.15\text{ km}$ is represented by a line $8.4\text{ cm}$ long. Find the scale in the form $1:n$. Another road is $11.2\text{ cm}$ long on the same drawing. Work out its actual length in kilometres.

A trapezium has parallel sides of lengths $7\text{ cm}$ and $13\text{ cm}$ and perpendicular height $6\text{ cm}$. Work out its area.

A cylinder has radius $4\text{ cm}$ and height $9\text{ cm}$. Work out its volume in terms of $\pi$.

A right triangular prism is $10\text{ cm}$ long. Its triangular cross-section has base $x\text{ cm}$ and perpendicular height $(x+2)\text{ cm}$. The volume is $600\text{ cm}^3$. Find $x$.

A circle has diameter $13\text{ cm}$. Write its circumference in terms of $\pi$.

A semicircle of diameter $10\text{ cm}$ is attached to one of the shorter sides of an $18\text{ cm}$ by $10\text{ cm}$ rectangle. The shared diameter is inside the shape. Work out the area and perimeter of the composite shape in terms of $\pi$.

A solid cone has radius $6\text{ cm}$, perpendicular height $8\text{ cm}$ and slant height $10\text{ cm}$. Work out its total surface area, including the circular base, and its volume. Give both answers in terms of $\pi$.

Work out the arc length of a quarter-circle with radius $8\text{ cm}$. Give the answer in terms of $\pi$.

A sector has radius $9\text{ cm}$ and arc length $7\pi\text{ cm}$. Work out the angle of the sector and its area in terms of $\pi$.

A shape is an annular sector with angle $144^\circ$, outer radius $12\text{ cm}$ and inner radius $7\text{ cm}$. Work out its area and perimeter in terms of $\pi$.

Triangle $ABC$ has $AB=5\text{ cm}$, $AC=7\text{ cm}$ and $\angle BAC=42^\circ$. Triangle $PQR$ has $PQ=5\text{ cm}$, $PR=7\text{ cm}$ and $\angle QPR=42^\circ$. State why the triangles are congruent.

Two similar shapes have corresponding lengths in the ratio smaller:larger $=3:5$. A side on the smaller shape is $12\text{ cm}$. Another side on the larger shape is $35\text{ cm}$. Work out the corresponding missing lengths.

Triangle $ABC$ has side lengths $8\text{ cm}$, $11\text{ cm}$ and $13\text{ cm}$. Similar triangle $DEF$ has longest side $19.5\text{ cm}$. Work out the other two side lengths of triangle $DEF$ and its perimeter.

A right-angled triangle has perpendicular sides $6\text{ cm}$ and $8\text{ cm}$. Work out the hypotenuse.

A straight ladder of length $7.2\text{ m}$ rests against a vertical wall. Its foot is $2.1\text{ m}$ from the wall. Work out the height reached by the ladder and the angle it makes with the ground. Give each answer to $1$ decimal place.

Isosceles triangle $ABC$ has $AB=AC=13\text{ cm}$ and $BC=10\text{ cm}$. Work out the perpendicular height from $A$ to $BC$ and angle $ABC$. Give the angle to $1$ decimal place.

Work out the exact value of $\sin30^\circ+\cos60^\circ$.

Work out the exact value of $2\sin45^\circ\cos45^\circ+\cos60^\circ$.

The hypotenuse of a right-angled triangle is $12\text{ cm}$ and one acute angle is $30^\circ$. Work out its exact area.

In triangle $ABC$, side $a=7\text{ cm}$, angle $A=30^\circ$ and angle $B=90^\circ$. Work out side $b$.

Two sides of a triangle are $7\text{ cm}$ and $10\text{ cm}$, and the included angle is $60^\circ$. Work out the exact length of the third side.

In triangle $ABC$, $A=35^\circ$, $a=8\text{ cm}$ and $b=11\text{ cm}$. Find all possible values of angles $B$ and $C$. Give each angle to $1$ decimal place.

Two sides of a triangle are $10\text{ cm}$ and $7\text{ cm}$, and their included angle is $30^\circ$. Work out the area.

A triangle has area $48\text{ cm}^2$. Two sides have lengths $12\text{ cm}$ and $x\text{ cm}$, and their included angle is $30^\circ$. Work out $x$.

Two sides of a triangle are $10\text{ cm}$ and $12\text{ cm}$. Its area is $30\sqrt3\text{ cm}^2$. Find both possible values of the included angle.

Point $A=(-3,4)$ is translated to $A'=(2,-1)$. Describe the translation as a vector.

Triangle $ABC$ has vertices $A=(1,-2)$, $B=(5,1)$ and $C=(2,4)$. It is translated by $\binom{-4}{3}$. Write the coordinates of the three image vertices.

A translation maps $P=(-6,4)$ to $P'=(1,-2)$. The same translation maps $Q=(3,k)$ to $Q'=(10,5)$. Find $k$ and describe the translation as a vector.

Work out $\binom{3}{-5}+\binom{-7}{2}$.

Points $A$ and $B$ have position vectors $\mathbf{a}=\binom{2}{-1}$ and $\mathbf{b}=\binom{8}{5}$. Point $M$ is the midpoint of $AB$. Find the position vector of $M$ and the vector $\overrightarrow{AM}$.

A walker starts at $P=(-3,2)$. The walker makes successive displacements $\binom{4}{3}$, $\binom{-1}{5}$ and $-2\binom{1}{-2}$. Work out the resultant displacement, the walker's finishing coordinates and the vector that would take the walker directly back to $P$.