P4 Geometry

64 practice problems for the State Matura in mathematics, with solutions.

Solve the problem in your notebook first, then open the solution. For worked problems, the points for each step are also given.

Problem 1 Multiple choice

The number of diagonals of a regular hexagon is:

A. 8

B. 9

C. 12

D. 15

Show solution

B. 9
$\frac{n(n-3)}{2}=\frac{6\cdot3}{2}=9$

Problem 2 Multiple choice

An isosceles triangle with leg 13 has height 12 to the base. The radius of the circumscribed circle is:

A. $\frac{13}{12}$

B. $4$

C. $\frac{169}{24}$

D. 5

Show solution

\frac{169}{24}

Problem 3 Short answer

Using the angles $α, β$ ( $β > α$ ) of the right triangle ABC, express the angle between the altitude and the median drawn from vertex C.

Show solution

$δ = β - α$

Problem 4 Short answer

What is the angle that is eight times its complement?

The diagonal of a square with side $a = 3$ is___________

Show solution

$\alpha=8(90^\circ-\alpha)\Rightarrow\alpha=80^\circ$

$3 \sqrt{2}$

Problem 5 Worked problem

When the number of sides of a regular polygon increases by 2, its central angle decreases by

6^{°}

. How many diagonals does the polygon have?

Show solution

Solution

$\frac{360^{°}}{n} - \frac{360^{°}}{n + 2} = 6^{°}$ 1 point

$n = 10$ 1 point

$\frac{n (n - 3)}{2} = \frac{10 (10 - 3)}{2} = 35$ 1 point

The number of diagonals is 35 1 point

Problem 6 Short answer

Find the perimeter of the right triangle ABC with hypotenuse $c = 5$ and $\cos α = \frac{4}{5}$ .

Show solution

Problem 7 Worked problem

The diagonals of the rhombus ABCD are in the ratio 3:4 and its area is 48

c m^{2}

. The length of a side of the rhombus is:

Show solution

Solution: $d_{1} : d_{2} = 3 : 4$ ,

$P = \frac{d_{1} d_{2}}{2} = \frac{3 k 4 k}{2} = 48$

$k = \sqrt{8}$ 1 point

$\begin{array}{l} d_{1} = 3 \sqrt{8} = 6 \sqrt{2} \\ d_{2} = 4 \sqrt{8} = 8 \sqrt{2} \end{array}$ 1 point

$d_{1}^{2} + d_{2}^{2} = 4 a^{2}$

$\begin{array}{l} 4 a^{2} = 72 + 128 \\ 4 a^{2} = 200 \end{array}$ $\begin{array}{l} a^{2} = 50 \\ a = \sqrt{50} \\ a = 5 \sqrt{2} \end{array}$ 1 point

Problem 8 Multiple choice

An isosceles trapezium has height $4 \sqrt{3}$ and leg 8. The sine of the acute angle of the trapezium is

A. $\frac{\sqrt{3}}{2}$

B. $\frac{\sqrt{2}}{2}$

C. 1

D. $\frac{1}{2}$

Show solution

\frac{\sqrt{3}}{2}

Problem 9 Worked problem

One side of a rectangle is 7

c m

, and the diagonal is 1

c m

longer than the other side. How long is that side?

Show solution

$d^{2} = a^{2} + b^{2}$ 1 point

${(x + 1)}^{2} = 7^{2} + x^{2}$ 1 point

$x = 24$ 1 point

Problem 10 Short answer

The longest chord of a circle is called the _________

Inscribed angles on the same arc are _________

Show solution

Diameter

Equal

Problem 11 Multiple choice

How many degrees is the angle between two neighbouring sides of a regular pentagon inscribed in a circle?

A. $100^{°}$

B. $72^{°}$

C. $108^{°}$

D. $50^{°}$

Show solution

108^{°}

Problem 12 Multiple choice

Find the height of a cuboid whose base edges are 3 and 5 and whose surface area is 190.

A. 10

B. 12

C. 15

D. 17

Show solution

A. 10

Problem 13 Multiple choice

Triangle ABC is inscribed in a circle so that side AB is a diameter. What kind of angle is the angle at vertex C?

A. right

B. obtuse

C. acute

D. Other

Show solution

A. Right

Problem 14 Short answer

Two sides of a triangle are 2 and 3 and the angle between them is $60^{°}$ . The length of the third side is:

Show solution

\sqrt{7}

Problem 15 Multiple choice

A student walked 1 km up a slope and rose 500 m. How many degrees is the angle of the slope?

A. $20^{°}$

B. $30^{°}$

C. $45^{°}$

D. $60^{°}$

Show solution

B. $30^{°}$

Problem 16 Multiple choice

An equilateral triangle has side 12. The radius of the circumscribed circle is

A. $3 \sqrt{3}$

B. $\frac{\sqrt{3}}{4}$

C. $2 \sqrt{3}$

D. $4 \sqrt{3}$

Show solution

4 \sqrt{3}

Problem 17 Short answer

The midline of a triangle is

4 c m

. The corresponding base of the triangle is _________

c m

Show solution

8 cm

Problem 18 Short answer

The area of a right triangle is $24 c m^{2}$ and one leg is $8 c m$ The length of the other leg is:

One angle of a triangle is $15^{°}$ larger than the second angle and $15^{°}$ smaller than the third. That angle is:

Show solution

$60^{°}$

Problem 19 Short answer

A cylinder and a cone have equal bases and heights. The volume of the cylinder is ______ times the volume of the cone.

A triangle and a parallelogram have equal bases and heights. The area of the parallelogram is ______ times the area of the triangle.

Show solution

three

two

Problem 20 Short answer

A pool of volume $100 m^{3}$ is filled by a pipe at $100 l / m i n$ and at the same time emptied at $1000 l / h$ . After how many hours will the pool be full?

Show solution

20 hours

Problem 21 Short answer

A pyramid with a square base has volume 32 and height 6. The side of the base is:

Show solution

Problem 22 Short answer

The diagonals of a rhombus are

6 c m

and

8 c m

. The perimeter of the rhombus is _____ cm

Show solution

20 cm
$a=\sqrt{3^2+4^2}=5,\ L=4a=20$

Problem 23 Worked problem

A square sheet of area

Q

is rolled into the lateral surface of a right circular cylinder. Find the area of the base of the cylinder.

Show solution

Writes that the perimeter of the base is $L = \sqrt{Q}$ 1 point

Gets the radius of the base $R = \frac{\sqrt{Q}}{2 π}$ 1 point

Calculates the area of the base $P = \frac{Q}{4 π}$ 1 point

Problem 24 Worked problem

The axial section of a cylinder is a square of area

12 c m^{2}

. Find the lateral area and the volume of the cylinder.

Show solution

From the axial section gets the radius of the cylinder $R = \sqrt{3}$ 1 point

Calculates the lateral area $P = 12 π c m^{2}$ 1 point

Gets the volume of the cylinder $V = 6 \sqrt{3} π c m^{3}$ 1 point

Problem 25 Short answer

How many denars does the material cost for a tent shaped like a regular square pyramid (without base) with base edge

3 m

and height

2 m

, if

1 m^{2}

costs 800 denars?

Show solution

Gets the slant height of the pyramid $h = 2.5 m$

Calculates the lateral area $15 m^{2}$

Gets the amount needed to buy the material $12\ 000$ den.

Problem 26 Worked problem

Find the volume V of a cone with radius

r = 7 c m

and lateral area

M = 175 π c m^{2}

Show solution

From $M = r π s$ gets $s = 25 c m$ 1 point

From $H^{2} = s^{2} - r^{2}$ gets $H = 24 c m$ 1 point

From $V = \frac{1}{3} π r^{2} H$ gets $V = 392 π$ 1 point

Problem 27 Worked problem

A right trapezium with bases

15 c m

and

9 c m

and area

96 c m^{2}

rotates about the longer base. Find the volume of the solid of revolution.

Show solution

Gets the radius of the solid of revolution (cylinder and cone) $R = 8$ 1 point

Gets the volume of the cylinder $V = 8^{2} π 9 = 576 π$ 1 point

Gets the volume of the cone $V = \frac{8^{2} π 6}{3} = 128 π$ 1 point

Gets the volume of the solid $V = 704 π c m^{3}$ 1 point

Problem 28 Short answer

If an inscribed angle is $32^{\circ}$ , then the central angle on the same arc is:

Show solution

$64^{\circ}$

Problem 29 Short answer

If 12 diagonals can be drawn from one vertex, the polygon has _________ sides

Show solution

15
$n-3=12\Rightarrow n=15$

Problem 30 Short answer

Two vectors with equal lengths and opposite directions are called __________ vectors.

Show solution

opposite

Problem 31 Short answer

Find the area of a rectangle with side 6 $cm$ and diagonal 10 $cm$

Show solution

$b = \sqrt{d^{2} - a^{2}} = \sqrt{10^{2} - 6^{2}} = \sqrt{100 - 36} = \sqrt{64} = 8$

$P = ab;P = 6 \cdot 8 = 48$

$P = 48\ cm^{2}$

Problem 32 Short answer

A rhombus with side 8 $cm$ and an angle of $45^{\circ}$ has area:

Show solution

$\sin 45^{\circ} = \frac{h}{8}\ $

$h = 8 \cdot \sin 45^{\circ} = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2}$

$P = ah = 8 \cdot 4\sqrt{2} = 32\sqrt{2}$

Problem 33 Short answer

The area of an isosceles triangle with base $a = 2$ and apex angle $60^\circ$

Show solution

The triangle is equilateral, so $P = \frac{a^{2}\sqrt{3}}{4} = \frac{4\sqrt{3}}{4} = \sqrt{3}$

Problem 34 Short answer

The area of a parallelogram is 54 $cm^{2}$ and its heights are 9 $cm$ and 12 $cm$. Its perimeter is:

Show solution

$54 = 9 \cdot a;a = 6$

$54 = 12 \cdot b;b = \frac{9}{2} = 4,5$

$P = 2 \cdot 6 + 2 \cdot 4,5 = 12 + 9 = 21$

Problem 35 Short answer

Find the radius of the circle inscribed in the triangle

with sides 26 $cm$, 28 $cm\ $and 30 $cm$.

Show solution

$s = \frac{26 + 28 + 30}{2} = 42$

$P = \sqrt{42(42 - 26)(42 - 28)(42 - 30)} = 336$

$r = \frac{P}{s} = \frac{336}{42} = 8$

Problem 36 Short answer

Find the area of an isosceles trapezium with bases 24 $cm$ and 14 $cm$ and leg 13 $cm$

Show solution

$P = \frac{a + b}{2} \cdot h$

$P = \frac{24 + 14}{2} \cdot h = 19h$

$h^{2} = 13^{2} - \left( \frac{a - b}{2} \right)^{2} = 13^{2} - 5^{2} = 144$ , $h = \sqrt{144} = 12,h = 12$

$P = 19 \cdot 12 = 228$

Problem 37 Short answer

The bases of a trapezium are 36 $cm$ and 12 $cm$, and the leg of length 7 $cm\ $makes an angle of $150^{\circ}$ with one base. Find the area of the trapezium.

Show solution

$h = 7 \cdot \sin 30^{\circ} = 7 \cdot \frac{1}{2} = \frac{7}{2}$

$P = \frac{a + b}{2} \cdot h = \frac{36 + 12}{2} \cdot \frac{7}{2} = 12 \cdot 7 = 84$

Problem 38 Short answer

The cross-section of an iron bar is a regular octagon. Find its area if the distance between its two farthest vertices is 12.

Show solution

\[P = 8\frac{6 \cdot 6 \cdot \sin 45^{\circ}}{2} = 8 \cdot 18 \cdot \frac{\sqrt{2}}{2} = 8 \cdot 9\sqrt{2} = 72\sqrt{2}\]

Problem 39 Short answer

Two circles have equal radii. If the radius of one is increased by 1 and the circumference of the other by 2$\pi$, which one will have the larger area?

Show solution

\[S = (R + 1)^{2}\pi\]

\[P = 2R\pi + 2\pi = 2\pi(R + 1)\]

\[S_{1} = (R + 1)^{2}\pi\]

Equal

Problem 40 Short answer

What is the radius of a circle if a sector with central angle $18^\circ$ has area 10?

Show solution

$P = \frac{R^{2}\pi \cdot \alpha}{360^{\circ}}\ $

$10 = \frac{R^{2} \cdot \pi \cdot 18^{\circ}}{360^{\circ}} = \frac{R^{2}\pi}{20}$

$\begin{array}{r} R^{2} = \frac{200}{\pi} \\ R = \sqrt{\frac{200}{\pi}} \end{array}$

Problem 41 Short answer

Two circles, each of radius 4 $cm$, intersect so that the common chord is 4 $cm$. Find the area of the region bounded by the two minor arcs

Show solution

\[P = 2\left( \frac{4^{2}\pi}{6} - \frac{4^{2}\sqrt{3}}{4} \right) = 16\left( \frac{\pi}{3} - \frac{\sqrt{3}}{2} \right)\]

Problem 42 Multiple choice

The perimeters of two similar triangles are in the ratio 3:5 and the sum of their areas is $340\ cm^{2}$. The area of the smaller triangle in square centimetres is:

A. 250

B. 90

C. 9

D. 127,5

Show solution

B. 90
The areas are in the ratio $9:25$: $9k+25k=340\Rightarrow k=10$, so $9k=90$.

Problem 43 Multiple choice

An isosceles triangle has base 6 and the medians to the legs are perpendicular. The area of the triangle is:

A. 12

B. 18

C. 24

D. 27

Show solution

D. 27

Problem 44 Multiple choice

A rhombus has diagonals 12 and 16. Joining the midpoints of its sides in order gives a new quadrilateral with equal diagonals. The area of the new quadrilateral is:

A. 24

B. 48

C. 72

D. 30

Show solution

B. 48
The new quadrilateral has half the area of the rhombus: $\frac12\cdot\frac{12\cdot16}{2}=48$.

Problem 45 Multiple choice

For the diagonals $d_{1}$ and $d_{2}$ of a rhombus with side $a$ and area $S$ it is true that:

A. $d_{1} + d_{2} = \sqrt{2S + 4a^{2}}$

B. $d_{1} + d_{2} = 2S + a^{2}$

C. $d_{1} + d_{2} = 2\sqrt{S + a^{2}}$

D. $d_{1} + d_{2} = 4S + 2a^{2}$

Show solution

C. $d_{1} + d_{2} = 2\sqrt{S + a^{2}}$

Problem 46 Short answer

A sphere of diameter $8\ cm$ is inscribed in a cylinder

A. The surface area of the cylinder is:__________$cm^{2}$

B. The volume of the cylinder is: ___________$cm^{3}$

Show solution

A.$96\pi$

Б. $128\pi$

Problem 47 Short answer

A cylindrical vessel of diameter 12 $cm$ is full of water. A stone is dropped in and some water spills out. When the stone is removed, the water level falls by 10 $cm$. The volume of the stone is_____________$cm^{3}$

Show solution

\[360\pi\]

$V=r^2\pi\cdot h=6^2\pi\cdot10=360\pi$

Problem 48 Short answer

The lateral area of a cone with height 12 and radius 5 is_______$cm^{2}$.

B. The volume of the cone is________$cm^{3}$

Show solution

A. $65\pi$

B. $100\pi$

Problem 49 Worked problem

The trapezium $ABCD(AB \parallel CD)\ $ has $AB = 24\ $ and $CD = 10$ and legs $AD = 15$ and $BC = 13$. Find the area of the trapezium. Draw a sketch.

Show solution

Draws a sketch Скица кон задачата (1 point)

writes Heron’s formula for the area of the triangle $\Delta BCE$ . (1 point)

Gets the area of the triangle $\Delta BCE$ $\ P =$84 (1 point)

Finds $h = 12$ (1 point)

Calculates the area of the trapezium $P = 204$. (1 point)

Problem 50 Worked problem

A cube is inscribed in a sphere. Find the ratio of their surface areas.

Show solution

Gets $a = \frac{2R}{\sqrt{3}}$ (1 point)

Writes $S_{t} = 4R^{2}\pi$

\[S_{k} = 6a^{2}\] (1 point)

Gets the ratio $\frac{S_{t}}{S_{k}} = \frac{\pi}{2}$ (1 point)

Problem 51 Worked problem

A hemisphere and a cone have a common base of radius $R$. The volumes of the cone and the hemisphere are equal. Find the lateral area of the cone. Draw a sketch.

Show solution

Draws a sketch (1 point)

Gets $h = 2R$ where $h$ is the height of the cone (1 point)

Gets the slant height of the cone $s = R\sqrt{5}$ (1 point)

Calculates the lateral area $P = \pi R^{2}\sqrt{5}$ (1 point)

Problem 52 Short answer

The area of a circle with circumference $8\pi\ cm$ is:______________$cm^{2}$

Show solution

\[16\pi\]

Problem 53 Short answer

The diagonal of a rectangle with area $48cm^{2}$ and side $6cm\ $is:

___________$cm$

Show solution

Problem 54 Short answer

The quadrilateral pyramid $ABCDM$ has a rectangular base $ABCD$ with $MD\bot ABCD$, $AB = 2,\ \ AD = \sqrt{5},\ \ DM = 2$. The volume of the pyramid is_________________

Show solution

\[\frac{4}{3}\sqrt{5}\]

Problem 55 Short answer

The diagonal of a regular square prism is $\sqrt{34}\ cm$ and the diagonal of its base is $3\sqrt{2}\ cm$.

A. The volume of the prism is _____________ $cm^{3}$

B. The surface area of the prism is _____________ $cm^{2}$

Show solution

A. $36$

B. 66

Problem 56 Short answer

A. The vectors $\overrightarrow{a} = (13,2),\ \overrightarrow{b} = (11,6)$ and $\ \overrightarrow{c} = (1,1)$ are given.

If $\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{u} = 4\overrightarrow{c} - \overrightarrow{u}$, then $\overrightarrow{u} = (\ \ \ \ \ \ \ \ \ \ \ ,\ \ \ \ \ \ \ \ \ \ )$.

B. A student runs at a constant speed of $6\ km/h$ against the wind. What is the student’s actual speed if the wind blows at a constant $3\ km/h$ ?

Show solution

A. (-10,-2)

B. $3km/h$

Problem 57 Short answer

Find the area of the section of a cone parallel to the base through the midpoint of the height, if the radius of the base is $20\ cm$.

The axial section of a cone is an equilateral triangle with side $a$ ; find the height of the cone.

\[H = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\]

Show solution

A. $100\pi cm^{2}$

B. $\frac{a\sqrt{3}}{2}$

Problem 58 Short answer

Find the radii of the spheres inscribed in and circumscribed about a cube with edge $a$.

The radius of the inscribed sphere is_____________

The radius of the circumscribed sphere is_____________

Show solution

$\frac{a}{2}$

B. $\frac{a\sqrt{3}}{2}$

Problem 59 Worked problem

The sides of the right triangle $\Delta ABC$ with perimeter 36 form an arithmetic progression ($AC < BC < AB$). Find the lengths of the sides of the triangle $ABC$.

Show solution

Writes the sides of the triangle in the form

\[b = a - d,a,a + d = c\] (1 point)

Gets $a = 12$ (1 point)

Gets $\ d = 3$ (1 point)

Gets the sides

$a = 12,\ b = 9,c = 15\ $ (1 point)

Problem 60 Worked problem

The base of the quadrilateral pyramid $ABCDM$ is a square with side $3a\ cm$. The lateral edge $DM$ is perpendicular to the plane of the base and has length $4a\ cm$ . Find the surface area of the pyramid.

Show solution

Writes the formula for the surface area of a pyramid

$P = B + M$

$M = M_{1} + M_{2} + M_{3} + M_{4}$

Gets $МА = MC = 5a$ (1 point)

Gets $B = 9a^{2}$ (1 point)

Gets $М_{1} = M_{2} = 6a^{2}$ (1 point)

Gets $М_{3} = M_{4} = \frac{15a^{2}}{2}$ (1 point)

Gets the surface area of the pyramid $P = 36a^{2}$ (1 point)

Problem 61 Multiple choice

The diagonal of a square with side $a = 2$ is:

A. 1

B. 2

C. $2\sqrt{2}$

D. $\sqrt{2}$

Show solution

C. $2\sqrt{2}$

Problem 62 Multiple choice

The midline of a triangle is $3cm$. The corresponding base of the triangle is:

A. $12cm$

B. $9cm$.

C. $6cm$.

D. $3cm$.

Show solution

C. $6cm$.

Problem 63 Multiple choice

The area of a right triangle is $24cm^{2}$ and one leg is $6cm$. The length of the other leg is:

A. $4cm$

B. $8cm$

C. $10cm$

D. $14cm$

Show solution

B. $8cm$

Problem 64 Multiple choice

The right triangle $ABC$ has the lengths shown in the picture. The value of $\overline{AD} = x\ $ is:

Скица кон задачата

A. $1cm$

B. $2cm$

C. $3cm$

D. $4cm$

Show solution

D. $4cm$
The height to the hypotenuse: $h^2=x\cdot8\Rightarrow32=8x\Rightarrow x=4$