P4 Geometry – Matura 2013

74 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.

Problems from the 2013 State Matura problem bank (basic and advanced level). The programme has changed since, so some topics are no longer part of the Matura.

Problem 1 Short answer

One side of a rectangle is 8 and the diagonal is 10. The perimeter of the rectangle is________________________
Find the area of a circular sector with radius 4 and central angle \(30^{\circ}\) .

Show solution

A. 28

B. \(\frac{4\pi }{3}\)

Problem 2 Short answer

A. Let the vector \(\vec{u}=(3,-2)\quad \) and \(\vec{v}=(6,-4)\) , then \(2 \vec{u}+3 \vec{v}=\)___________

Find the area of a regular hexagon with side 6.

Show solution

A. \((24, -16)\)

B. \(54\sqrt{3}\)

Problem 3 Short answer

Find the area of a triangle with sides 5, 6 and 9.

Show solution

A. \(10 \sqrt{2}\)

Problem 4 Short answer

The bases of a trapezium are 35 and 29 and its area is 256. Find the height of the trapezium.

Answer .________________

Show solution

Problem 5 Short answer

If the sides of a parallelogram are \(a=4 , b=6\) and the distance between its shorter sides is 5, find the distance between the longer sides.

Show solution

\(\frac{10}{3}\)

Problem 6 Multiple choice

Picture for the problem A line through the vertex \(D\) of a rhombus cuts the extensions of the sides \(B A\) and \(B C\) at the points \(E\) and \(F\) respectively. If \(A E=4\) and \(C F=9\) , find the length of a side of the rhombus.

A. 4

B. 6

C. 8

D. 10

Show solution

B. 6

Problem 7 Multiple choice

Picture for the problem A square with side \(y\) is drawn inside a square with side \(x\) . Find the ratio \(x\quad : y\) if the unshaded part of the big square has 8 times the area of the shaded part.

A. 3

B. \(\sqrt{8}\)

C. 9

D. 8

Show solution

A. 3

Problem 8 Multiple choice

Picture for the problem Two triangles, each with perimeter \(1 \,\text{dm}\) , are cut out of a square. What is the perimeter of the light figure that is left?

A. \(20 \,\text{cm}\)

B. \(2.5 \,\text{dm}\)

C. \(12 \,\text{cm}\)

D. \(3 \,\text{dm}\)

Show solution

A. \(20 \,\text{cm}\)

Problem 9 Multiple choice

Picture for the problem In the picture the ray \(B A \) is parallel to \(D E\) , the angle \(\angle B C D=30^{\circ}\) , the angle \(\angle C D E =140^{\circ}\) . Find the angle \(\angle C B A=? \) .

A. \(170^{\circ}\)

B. \(110^{\circ}\)

C. \(140^{\circ}\)

D. \(130^{\circ}\)

Show solution

D. \(130^{\circ}\)

Problem 10 Multiple choice

The picture shows the parallelogram \(A B C D\) , \(A E \) bisects \(\angle DAB\) , \(| A B |=8\,\text{cm}\) , \(| B C |=5\,\text{cm}\) , then \(| E C | =?\)

Picture for the problem

A. 1

B. 2

C. 3

D. 4

Show solution

C. 3

Problem 11 Multiple choice

The picture shows the right trapezium \(A B C D\) , \(| A D |=| D C | \) , \(| B C |=8\) , \(\angle B=45^{\circ}\) . Find the area of the trapezium.

Picture for the problem

A. 24

B. 36

C. 48

D. 64

Show solution

C. 48

Problem 12 Multiple choice

Drawing for the problem

In the picture \(P B \) and \(P C \) are tangents to the circle, \(\angle P=80^{\circ}\) , then \(\angle B A C=?\) .

A. \(50^{\circ}\)

B. \(60^{\circ}\)

C. \(70^{\circ}\)

D. \(80^{\circ}\)

Show solution

A. \(50^{\circ}\)

Problem 13 Worked problem

Drawing for the problem

In the picture \(| A B |= | B C |=| B B ' | \) , \(| P A |=| P A ' | \) . Find \(|PC'|=?\)

Show solution

Finds \(| A' C ' |=8\sqrt{2}\) (1 point)

Writes Pythagoras’ theorem for the triangle \(\triangle A 'C ' P\) . (1 point)

Calculates \(| P C ' |=12\) (1 point)

Problem 14 Worked problem

Drawing for the problem

The picture shows a right regular pyramid with a square base, i.e. \(| A B |=| B C |=| D C |=| D A |\) , \(| A B |=32 \,\text{cm}\) . The area of the triangle \(\triangle BPC\) is \(320 {\,\text{cm}}^{2}\) . Find the volume of the pyramid.

Show solution

Labels the elements in the picture. (1 point)

Finds the slant height of the pyramid \(h=20 \,\text{cm}\) (1 point)

Finds the height of the pyramid \(H=12 \,\text{cm}\) (1 point)

Calculates the volume of the pyramid \(V=4096 {\,\text{cm}}^{3}\) . (1 point)

Problem 15 Worked problem

Drawing for the problem

The picture shows a right triangle with \(| A B |= | B C | \) , \(| E C |\quad :\quad | B C |=1 : 3 \) ; the area of the triangle \(AEC\) is \(24 {\,\text{cm}}^{2}\) . Find the length of the segment \(| A B |=?\)

Show solution

Concludes that \(P(\triangle ABC)=3 P(\triangle AEC)\) , gets \(P(\triangle ABC)=72{\,\text{cm}}^{2}\) (1 point)

Writes \(P(\triangle ABC)=\frac{{x}^{2}}{2}=72\) (1 point)

Finds \(x=| A B |=12 \,\text{cm}\) (1 point)

Problem 16 Worked problem

A circle of radius 2 is inscribed in an isosceles trapezium of area 20. Find the sides of the trapezium.

Show solution

From \(P=\frac{a+b}{2} \cdot h\) gets \(a+b=10\) , \(h=4\) (1 point)

Since it is a tangential quadrilateral, writes \(a+b=2c\) , \(c=5\) . (1 point)

From \(c=5\quad h=4 \) gets \(a-b=6\) (1 point)

From \(a-b=6\) \(a+b=10\) gets \(a=8 b=2\) (1 point)

Problem 17 Worked problem

The isosceles triangle \(\triangle ABC\) has base \(A B=8\sqrt{5}\) and leg \(A C=12\) . Find the radius of the circle circumscribed about the triangle.

Show solution

Finds the height of the triangle \(h=8\) (1 point)

Finds the area of the triangle \(P=32\sqrt{5}\) (1 point)

From \(R=\frac{abc}{4P}\) gets \(R=9\) (1 point)

Problem 18 Short answer

Fill in the table

	Cube	Regular square pyramid
A. Number of edges
B. Number of faces

Show solution

	Cube	Regular square pyramid
A. Number of edges	12	8
B. Number of faces	6	5

Problem 19 Short answer

Drawing for the problem

The area of the shaded part of the picture is _________________ \({\,\text{cm}}^{2}\) .

Show solution

\((100-12.5 \pi ){\,\text{cm}}^{2}\)

Problem 20 Short answer

A. A circle has radius \(12\,\text{cm}\) . Find the area of the square in which the circle is inscribed.

B. The circumference of a circle with radius \(r\) is given by the formula___________________

Show solution

A. \(576{\,\text{cm}}^{2}\)

B. \(L=2r\pi \)

Problem 21 Short answer

Picture for the problem A right triangle with leg \(4\,\text{cm}\) is removed from a square with side \(7\,\text{cm}\) (as in the picture). Find the perimeter of the resulting pentagon.

Answer .________________

Show solution

\(20+4\sqrt{2}\)

Problem 22 Short answer

In an isosceles right triangle the hypotenuse is \(10\sqrt{2} \,\text{cm}\) . The area of the triangle is_______________ \({\,\text{cm}}^{2}\) .

Show solution

\(50\)

Problem 23 Short answer

From the picture find: \(x+y=\) _____________

Picture for the problem

In the picture the line \(a\) is parallel to the line \(b\) ; find the angle \(x=\) ________________.

Drawing for the problem

Show solution

\(155^{\circ}\)
\(130^{\circ}\)

Problem 24 Short answer

Picture for the problem

Drawing for the problem

The area of the parallelogram in the picture is _____________

B. The volume of a cuboid with edges 8, 4, 2 is _____________.

Show solution

B. 64

Problem 25 Short answer

A. The area of a square with perimeter 12 is ___________

B. The area of a circle with radius 2 is ___________

Show solution

A. 9

B. \({2}^{2} \pi \)

Problem 26 Short answer

Find the surface area of the cylinder in the picture.

(A cylinder with radius \(r=6\,\text{cm}\) and height \(H=7\,\text{cm}\).)

Show solution

\(156\pi\,\text{cm}^2\)
\(P=2r\pi(r+H)=2\cdot6\pi\cdot13=156\pi\)

Problem 27 Short answer

Find the volume of the pyramid in the picture

(A regular square pyramid with height \(H=16\,\text{cm}\) and base edge \(a=24\,\text{cm}\).)

Show solution

3072 \(\,\text{cm}^3\)

Problem 28 Multiple choice

Given \((-1,4)+\vec{v}=(4,5)\) , then \(\vec{v}\) is equal to:

A. (3,9)

B. (-5,1)

C. (5,1)

D. (9,3)

Show solution

C. (5,1)

Problem 29 Multiple choice

Let \(\vec{u}=\vec{i}-5\vec{j}\) and \(\vec{v}=2\vec{i}+3\vec{j}\) be given. Find \(3\vec{u}+2\vec{v}\)

A. \(3\vec{i}-2\vec{j}\)

B. \(5\vec{i}-\vec{j}\)

C. \(7\vec{i}-9\vec{j}\)

D. \(2\vec{i}-3\vec{j}\)

Show solution

C. \(7\vec{i}-9\vec{j}\)

Problem 30 Multiple choice

A plane has no

A. thickness

B. length

C. width

D. surface

Show solution

A. thickness

Problem 31 Multiple choice

According to the picture Picture for the problem

A. A, B and E are collinear points.

B. \(s \cap d=\{B\}\)

C. \(C \in s\)

D. D, B, E are non-collinear points

Show solution

B. \(s \cap d=\{B\}\)

Problem 32 Multiple choice

How many lines are determined by five points, no three of which are collinear?

A. 15

B. 12

C. 10

D. 9

Show solution

C. 10

Problem 33 Multiple choice

What is the sum of the exterior angles of a triangle?

A. \(90^{\circ}\)

B. \(180^{\circ}\)

C. \(270^{\circ}\)

D. \(360^{\circ}\)

Show solution

D. \(360^{\circ}\)

Problem 34 Multiple choice

The angle \(\frac{3\pi}{4}\,\text{rad}\) in degrees is:

A. \(90^\circ\)

B. \(135^\circ\)

C. \(150^\circ\)

D. \(210^\circ\)

Show solution

B. \(135^\circ\)

Problem 35 Multiple choice

If 10 diagonals can be drawn from one vertex of a polygon, how many sides does the polygon have:

A. 10

B. 12

C. 13

D. 15

Show solution

C. 13

Problem 36 Multiple choice

Each interior angle of a regular hexagon is:

A. \(108^{\circ}\)

B. \(120^{\circ}\)

C. \(136^{\circ}\)

D. \(100^{\circ}\)

Show solution

B. \(120^{\circ}\)

Problem 37 Multiple choice

In the picture the line a is parallel to the line b. Find the angle x.

Drawing for the problem

A. \(95^\circ\)

B. \(105^\circ\)

C. \(110^\circ\)

D. \(100^\circ\)

Show solution

D. \(100^\circ\)

Problem 38 Multiple choice

Which segments in the picture are equal:

Picture for the problem

A. AC and BD

B. OE and OA

C. AB and CD

D. OB and OC

Show solution

D. OB and OC

Problem 39 Multiple choice

Find the volume of a prism whose base is an equilateral triangle with side 4 and whose length is 6.

Picture for the problem

A. 24

B. \(16\sqrt{3}\)

C. \(24\sqrt{3}\)

D. 48

Show solution

C. \(24\sqrt{3}\)

Problem 40 Multiple choice

Which formula gives the surface area of a cylinder with one open end and one closed end?

A. \(2r\pi H+2r^2\pi\)

B. \(2r\pi H+r^2\pi\)

C. \(2r\pi H-r^2\pi\)

D. \(2r\pi H\)

Show solution

B. \(2r\pi H+r^2\pi\)

Problem 41 Multiple choice

The areas of the discs bounded by two concentric circles are in the ratio 2:1. The outer circle has radius \(8\). Find the radius of the inner circle.

Drawing for the problem

A. 5

B. \(4\sqrt{2}\)

C. 3

D. \(2\sqrt{3}\)

Show solution

B. \(4\sqrt{2}\)

Problem 42 Multiple choice

The lateral surface of a cone is a:

A. circular sector

B. circular segment

C. annulus

D. another answer

Show solution

A. circular sector

Problem 43 Worked problem

A cylindrical pot is \(\frac{8}{9}\) full of water. How many litres of water are in the pot if it is \(32\,\text{cm}\) high and its diameter is \(30\,\text{cm}\) ? Draw a sketch.

Show solution

Draws a sketch (1 point)

Writes the formula \(V=\frac{8}{9} \cdot \pi \cdot {1.5}^{2} \cdot 3.2\) (1 point)

Calculates \(V=20{,}106 {\,\text{dm}}^{3} (20106 {\,\text{cm}}^{3})\) (1 point)

Concludes: about 20 (20.1) litres of water. (1 point)

Problem 44 Worked problem

A regular square pyramid has volume \(400 {\,\text{cm}}^{3}\) and height \(12\,\text{cm}\) .

Draw a sketch and find the base edge.
Find the surface area of the pyramid.

Show solution

Draws a sketch and writes the volume formula

\(V=\frac{{a}^{2}H}{3}\) (1 point)

Finds \(a=10\,\text{cm}\) (1 point)

Finds the slant height of the pyramid \(h=13\,\text{cm}\) (1 point)

Calculates the surface area of the pyramid \(P=360 {\,\text{cm}}^{2}\) (1 point)

Problem 45 Worked problem

Find the length of wire needed to make a model of a cube with edge \(10\,\text{cm}\) including one space diagonal. Draw a sketch.

Show solution

Draws a sketch (1 point)

Calculates \(l=12a\) (1 point)

Writes

\(D=a \sqrt{3}\) (1 point)

Calculates \(L=l+D=137{,}3 \,\text{cm}\) (1 point)

Problem 46 Worked problem

Drawing for the problem

The sketch shows the axial section of a cylinder. Find the surface area of the cylinder.

Picture for the problem

Show solution

Writes \(r=2\,\text{cm}\quad H=10 \,\text{cm}\) (1 point)

Writes the formula for the surface area

\(P=2\pi \cdot 2(2+10)\quad \) (1 point)

Finds \(P=48 \pi {\,\text{cm}}^{2}\)

The problem is also accepted if solved with

\(r=5 , H=4, P=90\pi {\,\text{cm}}^{2}\) . (1 point)

Problem 47 Worked problem

Balls of diameter \(0.4\,\text{cm}\) are made from a gold bar shaped like a cuboid with dimensions \(10\,\text{cm}\) , \(5\,\text{cm}\) and \(4\,\text{cm}\) . What is the largest number of balls that can be made?

Show solution

Writes the formula for the volume of a ball \({v}_{t}=\frac{4\pi {r}^{3}}{3}\) (1 point)

Writes the formula for the volume of the cuboid and gets \({V}_{k}=abc=200{\,\text{cm}}^{3}\) (1 point)

Writes the quotient \(n={V}_{k}:{V}_{t}\) (1 point)

Concludes that \(n=\frac{200}{0{,}0335}\approx 5968\) balls can be made. (1 point)

Problem 48 Worked problem

A right cone has radius \(3\,\text{cm}\) and slant height \(5\,\text{cm}\) . Draw a sketch, mark the axial section and find its area.

Show solution

Picture for the problem

Draws a sketch and labels the needed elements (1 point)

Calculates \(H=\sqrt{{s}^{2}-{r}^{2}}=4\,\text{cm}\) (1 point)

Calculates the area of the axial section \(P=12{\,\text{cm}}^{2}\) . (1 point)

Problem 49 Worked problem

A right triangle is given. Find the sides of the triangle.

Picture for the problem

Show solution

Sets up the equation

\({(x+3)}^{2}={x}^{2}+{(x-3)}^{2}\) (1 point)

Gets \({x}_{1}=0\quad {x}_{2}=12\) (1 point)

Concludes: the sides of the triangle are 9, 12 and 15 (1 point)

Problem 50 Worked problem

The diagonals of a rhombus are \({d}_{1}=16 \,\text{cm} \) and \({d}_{2}=12\,\text{cm}\) . Draw a sketch and find the perimeter of the rhombus.

Show solution

Drawing for the problem

Draws a sketch and labels the needed

elements.

Picture for the problem (1 point)

Gets \(a=10\,\text{cm}\) (1 point)

Calculates the perimeter \(L=40\,\text{cm}\) (1 point)

Problem 51 Worked problem

Find the area of the shaded part of the picture.

Picture for the problem

Show solution

Calculates the area of the square \({S}_{1}=100{\,\text{cm}}^{2}\) (1 point)

Calculates the area of the triangle

\({S}_{2}=\frac{{a}^{2}\sqrt{3}}{4}\) (1 point)

Gets \(S={S}_{1}-{S}_{2}=(100-25 \sqrt{3} ){\,\text{cm}}^{2}\) (1 point)

Problem 52 Worked problem

Find the volume of a ball with surface area \(P=196\pi\).

Show solution

Writes the formula for the surface area of a ball

\(P=4R^2\pi\)

Finds the radius of the ball \(R=7\) (1 point)

Writes the formula for the volume of a ball

\(V=\frac{4}{3}R^3\pi\) (1 point)

Calculates the volume of the ball \(V=\frac{1372}{3}\pi\) (1 point)

Problem 53 Short answer

Find the unknown lengths \(x\) and \(y\) in the sketch.

Picture for the problem

Answer:

Drawing for the problem

A. \(x=\) ___________

B. \(y=\) ___________

Show solution

A. \(\frac{30}{7} (4.3)\)

B. \(\frac{15}{2} (7.5)\)

Problem 54 Short answer

Find the angles \(x\) and \(y\) shown in the picture.

Picture for the problem

A. \(x=\) __________

B. \(y=\) __________

Show solution

A. \(x=70^{\circ}25' (70.42^{\circ})\)

B. \(y=109^{\circ}35' (109,58^{\circ})\)

Problem 55 Short answer

Find the perimeter of the figure in the picture.

Picture for the problem

Answer: \(L=\)

Show solution

A. \(L=3\pi +30\)

Problem 56 Short answer

The triangle \(ABC\) is given. The segment EF is parallel to the side AB. Find the angle \(\gamma \) .

Picture for the problem

Answer: \(\gamma =\) ____________

Show solution

\(110^{\circ}\)

Problem 57 Short answer

The circles have radii \(R=8 m\) and \(r=4 m\) . Find the area of the shaded part.

Picture for the problem

Show solution

\(48 \pi \)

Problem 58 Short answer

Find the volume of a ball with surface area \(P=196\pi\).

2. The median to one of the sides of an isosceles right triangle has length 5. Find the area of the triangle.

Show solution

A. \(\frac{1372}{3}\pi\)

B. 10

Problem 59 Multiple choice

In an isosceles triangle the base is 12 and the height to the base is 8. The perimeter of the triangle is:

A. 22

B. 24

C. 28

D. 32

Show solution

D. 32

Problem 60 Multiple choice

In an isosceles trapezium the bases are 6 and 16 and the leg is 13. The area of the trapezium is:

A. 110

B. 120

C. 132

D. 143

Show solution

C. 132

Problem 61 Multiple choice

A square is circumscribed about a disc of area \(4\pi \) . The radius of the circle circumscribed about the square is:

A. 2

B. \(\sqrt{8}\)

C. 4

D. \(4\sqrt{2}\)

Show solution

B. \(\sqrt{8}\)

Problem 62 Multiple choice

What is the radius of the circle circumscribed about a right triangle with a side of 5 and the angle opposite it \(30^{\circ}\) ?

A. 1

B. 2

C. 3

D. 5

Show solution

D. 5

Problem 63 Multiple choice

The sketch shows a circle and its chords \(\overline{AB} \) and \(\overline{CD}\) . The lengths in the sketch are: \(| D E |=7, | B E |=6, | C E |=3, | A E |=x\) . What is \(x\) ?

Picture for the problem

A. 2

B. 2.7

C. 3.5

D. 4

Show solution

C. 3.5

Problem 64 Multiple choice

What is the surface area of a regular triangular pyramid (tetrahedron) with all edges of length \(3 \,\text{cm}\) ?

A. \(\frac{9\sqrt{3}}{2} {\,\text{cm}}^{2}\)

B. \(9\sqrt{3} {\,\text{cm}}^{2}\)

C. \(\frac{27\sqrt{3}}{4} {\,\text{cm}}^{2}\)

D. \(27\sqrt{3} {\,\text{cm}}^{2}\)

Show solution

B. \(9\sqrt{3} {\,\text{cm}}^{2}\)

Problem 65 Multiple choice

Find the radius of the circle in the sketch:

Picture for the problem

A. \(\sqrt{50}\)

B. 8

C. \(\sqrt{113}\)

D. 25

Show solution

D. 25

Problem 66 Multiple choice

The perimeter of a rhombus with diagonals 12 and 16 is?

A. 20

B. 28

C. 32

D. 40

Show solution

D. 40

Problem 67 Multiple choice

The surface area of a right circular cone with base radius \(r=4\) and height \(H=3\) is:

A. \(12\pi \)

B. \(16\pi \)

C. \(24\pi \)

D. \(36\pi \)

Show solution

D. \(36\pi \)

Problem 68 Multiple choice

In the triangle \(ABC\) the point \(T\) is the intersection of the medians. Find the area of the triangle \(ABC\) if the area of the triangle \(BCT\) is \(12\) .

Picture for the problem

A. 24

B. 36

C. 18

D. 30

Show solution

B. 36

Problem 69 Worked problem

A regular square pyramid has volume \(400 {\,\text{cm}}^{3}\) and height \(12 \,\text{cm}\) .

Draw a sketch and find the base edge and the surface area of the pyramid.

Show solution

Draws a sketch with the elements needed for the problem (1 point)

From the formula \(V=\frac{{a}^{2}\cdot H}{3}=400\) , gets \(a=10 \,\text{cm}\) (1 point)

Calculates the slant height \(h=13 \,\text{cm}\) (1 point)

Calculates the surface area of the pyramid \(P=360 {\,\text{cm}}^{2}\) (1 point)

Problem 70 Worked problem

The isosceles trapezium \(ABCD\) has bases \(| A B |=a=10\,\text{cm}, | C D |=c=6\,\text{cm}\) and the angle \(\angle DAB=60^{\circ}\) .

A. Draw a sketch of the trapezium and find its area

B. Find the length of the diagonal \(B D.\)

Show solution

Draws a sketch and labels the needed elements:

Picture for the problem (1 point)

Calculates the height of the trapezium

\(h=\frac{a-c}{2} \cdot \operatorname{tg} \alpha =2\sqrt{3}\) (1 point)

Calculates the area of the trapezium

\(P=\frac{a+b}{2} \cdot h=16 \sqrt{3}\) (1 point)

Calculates the diagonal

\(|BD|=\sqrt{76}=2\sqrt{19}\) (1 point)

Problem 71 Worked problem

A rectangular yard is \(10 m\) long and \(6 m\) wide. The owner increased the length by 20% and decreased the width by 15%. By how many square metres ( \({m}^{2}\) ) does the area of the yard change?

Show solution

Calculates the area of the yard

\({P}_{1}=60 {m}^{2}\) (1 point)

Calculates the new length of the side \(12 m\) (1 point)

Calculates the new width of the side \(5.1 m\) (1 point)

Calculates the new area \({S}_{2}=61.2 {m}^{2}\) and gets the difference \(1.2 {m}^{2}\) . (1 point)

Problem 72 Worked problem

A rectangle with sides \(10 \,\text{cm}\) and \(4 \,\text{cm}\) rotates \(360^{\circ}\) about its longer side. Find the surface area of the resulting cylinder.

Show solution

Concludes \(a=H\quad b=r\)

\(a=10 \,\text{cm}\) , \(b=4 \,\text{cm}\) .

Picture for the problem (1 point)

Writes the formula for the surface area of a cylinder \(P=2\pi r(r+H)\) (1 point)

Calculates

\(P=2 \cdot 4 \cdot \pi \cdot (4+10)=112 \pi \) (1 point)

Problem 73 Worked problem

In an isosceles (right) triangle the hypotenuse is \(c=10 \sqrt{2} \,\text{cm}\) . Find the area of the triangle.

Show solution

Calculates the side \(a=10 \,\text{cm}\) (1 point)

Writes the formula for the area \(P=\frac{{a}^{2}}{2}\) (1 point)

Gets \(P=50 {\,\text{cm}}^{2}\) (1 point)

Problem 74 Worked problem

In the parallelogram \(ABCD \), \(\overline{AK} \perp \overline{BH}\) . The area of the parallelogram is \(72 {\,\text{cm}}^{2}\) , \(| B H |=6\,\text{cm}\) and \(K\) is any point of \( \overline{DC}\) ; find the length of the segment \(\overline{AK}\) .
Picture for the problem

Show solution

Concludes \(P(ABK)=\frac{1}{2} P(ABCD)=\frac{72}{2}=36\) (1 point)

Writes the formula for the area of the triangle \(P(ABK)=\frac{1}{2} | B H |\cdot | A K |\) (1 point)

Substitutes into the previous formula and gets \(| A K |=12 \,\text{cm}\) (1 point)