PLANE

By Marsigit

Basic Competency
To identify the properties of triangle in term of its sides and angles
To identify the properties of rectangle, square, trapezium, parallelogram, rhombus and kite
To determine the perimeter and area of triangle and rectangle and apply them to solve mathematical problems.
To construct triangle, altitude line, interior angle bisector line, median line, and perpendicular bisector line.

Two dimensional shapes such as triangle, rectangle, and square are easily found in your surroundings. For example, the front side of the roof of a building can be seen as a triangle shape and tennis court can be in the form of rectangular. There are still other two dimensional shapes, for example, parallelogram, trapezium, kite and rhombus.
Can you find the examples of each of those geometrical shapes from your surroundings.

What will you learn in this chapter?
Triangle
Rectangle
Triangle
Understanding Triangle
Triangle is a plane figure with three straight sides. You can construct a triangle by following methods:
Connecting Three Non-Collinear Points to Each Other.
Consider the following figure.
[..gambar 3 titik dan segitiga.. ]

Points A, B and C above do not lie in one straight line (non-collinear). If you connect those three points each other, you will get a triangle ABC.

Note:
Triangle is usually symbolized with “ ∆” . So, you can notate triangle ABC as ∆ABC.

Connecting The Ends Points of A Line Segment to A Certain Point Which not Lies In The Segment Line.

[.......gambar ...]

Let ¯AB is a line segment. Construct point C not in the line segment ¯AB . If you connect point C to the ends of line segment ¯AB you will get a triangle as it is shown in the above figure.

Dividing a Square or Rectangle Through Its Diagonal

Consider the following square ABCD and rectangle EFGH

If you divide the square or the rectangle through its diagonals, you will get two congruence triangles.

[gambar ]
Classifying Triangles

Three types of triangles are distinguished as follows:
Type of triangles in term of the measure of their angles
Type of triangles in term of the length of their sides
Type of triangles in term of the length of their sides and the measure of their angles

Classifying Triangles by Angles

In term of the measure of their angles there are three types of triangles i.e. acute triangle, right triangle and obtuse triangle.
A triangle is acute if all three of its angles are acute
A triangle is obtuse if one of their angles is obtuse
A triangle is right if one of their angles is right

[Gambar 3 buah Segitiga]

Note:
An angle A is right if A = 90°
An angle A is acute if 0° < A < 90°;
An angle A is obtuse if 90° < A < 180°.

Classifying Triangles by Sides

In term of the length of their sides there are three types of triangles i.e. isosceles triangle, equilateral triangle, and scalene triangle.
A triangle is isosceles if two sides are equal
A triangle is equilateral if all three sides are equal
A triangle is scalene if no two of its sides are equal in length

[gambar 3 buah segitiga]
Classifying Triangles by Angles and Sides
In term of the length of their sides and the measure of their angles, there are some types of triangles as follows:
A right isosceles triangle is a triangle in which one of their angles is right and two sides are equal
An acute isosceles triangle is a triangle in which all three of its angles are acute and two sides are equal
An obtuse isosceles triangle is a triangle in which one of their angles is obtuse and two sides are equal
A right scalene triangle is a triangle in which one of their angles is right and no two of its sides are equal in length
An acute scalene triangle is a triangle in which all three of its angles are acute and no two of its sides are equal in length
An obtuse scalene triangle is a triangle in which one of their angles is obtuse and no two of its sides are equal in length

Example
Describe the type of the following triangles in term of the measure of their angles


[gambar 3 buah segitiga]


Solution:
The first triangle is right
The second triangle is obtuse
The third triangle is acute

Exercise
Describe the type of the following triangles in term of the measure of their angles

[Gambar 3 buah segitiga]

Consider the following square!
Construct both diagonals of that square. Then describe four triangles that you found!
[Gambar persegi]
Construct three type of triangles as follows:
An acute isosceles triangle
An obtuse triangle
An equilateral triangle
An obtuse isosceles triangle
3. Constructing Triangles
a. Constructing Equilateral Triangles
The series of diagrams below shows the different stages of construction.
Let, you will construct Equilateral Triangle ∆ABC with the length of its side is 4 cm.
Construct 4 cm line segment ¯AB using pencil and ruler
At point A, set compass to a wide of line segment ¯AB to produce an arch
At point B, set compass to a wide of line segment ¯AB to produce other arch. You find that the second arch intersects with the first arch. Call the intersection point as C.
Connect point C to point A and to point B

[Gambar Lukisan Segitiga]

Exercise
Construct the following equilateral triangles!
∆ABC with the length of the sides is 5 cm
∆PQR with the length of the sides is 6.5 cm
∆ABC with the length of the sides is 4.5 cm
Construct equilateral triangle ABC in which the line segment ¯AB is as a base in the following positions!

[Gambar alas segitiga]


b. Constructing Isosceles Triangles
Followings are the stages of constructing isosceles triangles
Let you will construct isosceles triangle PQR with the length of the sides PR = QR =
cm.
Construct line segment ¯PQ . The length of this segment is arbitrary, i.e. PQ = 2 cm
At point P, set compass to a wide of 3.5 cm to produce an arch.
At point Q, set compass to a wide of 3.5 cm to produce other arch. You find that the second arch intersects with the first arch. Call the intersection point as R.
Connect point R to point P and to point Q
[Gambar lukisan segitiga samakaki]
Note:
A triangle is isosceles if two sides are equal
Exercise
Construct the following isosceles triangles!
∆ABC, AB = AC = 6 cm
∆KLM, KL = KM = 5 cm
b. Constructing Scalene Triangles
If you want to construct any triangle, make sure that the length of the sides should comply with triangular inequalities, i.e. the length of each side is less than the sum of the lengths of the other two sides.
Let, the length of AB, BC, and AC are consecutively c unit of length, a unit of length and b unit of length. If ABC is a triangle, it should comply with the following triangular inequalities:
a + b > c,
a + c > b, and
b + c > a.
In other words, you can construct a scalene triangle if the length of each side is less than the sum of the lengths of the other two sides.

Followings are the stages of constructing isosceles triangles
Let, you are asked to construct ∆ABC, AB = 4 cm, AC = 6 cm, and BC = 5 cm.
Construct 6 cm length of line segment ¯AC
Construct the side of AB by setting compass at point A to a wide of 4 cm to produce an arch.
Construct the side of BC by setting compass at point B to a wide of 5 cm to produce another arch. Call the intersection point of two arches as B.
Connect point B to point A and point C.


. The length of this segment is arbitrary, i.e. PQ = 2 cm
At point P, set compass to a wide of 3.5 cm to produce an arch.
At point Q, set compass to a wide of 3.5 cm to produce other arch. You find that the second arch intersects with the first arch. Call the intersection point as R.
Connect point R to point P and to point Q

[Gambar lukisan segitiga sembarang]

Exercise
Is it possible for you to construct the triangles with the length of the sides as follows? If it is possible, construct them!
PQ = 3,5 cm; PR = 5 cm; and QR = 4 cm
PQ = 3,5 cm; PR = 6 cm; and QR = 8 cm
PQ = 3,5 cm; PR = 6 cm; and QR = 5 cm
AB = 4 cm; BC = 7 cm; and AC = 5 cm
AB = 5 cm; BC = 9 cm; and A = 8 cm

4.Lines inTriangles

There are specific lines in a certain triangle:

Altitude of triangle
Bisector of triangle
Median of triangle and perpendicular bisector of triangle

Altitude of Triangles

The altitude of a triangle is the line passing through one vertex of the triangle, perpendicular to the line including the side opposite this angle.

The stages of constructing altitude of triangle:
Construct triangle ABC
At point A, set compass to a certain wide so that the resulting arch intersects with the opposite side of angle A i.e. side BC. Call the intersection point as P and Q.
At point P, set compass to a similar wide with that of procedure 2 to produce an arch.
At point Q, set compass to a similar wide with that of procedure 3 to produce another arch. Call the intersection point of the arches as M.
Connect point A to point M.
Line segment AM intersects with line segment BC at point D.
Line segment AD is called the altitude of triangle ABC.

[Gambar lukisan garis tinggi)

Exercise
Construct the altitude of the following triangles!

[Gambar 3 buah segitiga]

Bisector of Triangles

Bisector of a triangle is the line that divides the certain angle of a given triangle into smaller angles of equal size.

The stages of constructing bisector of triangle:
Construct a triangle ABC
At point A, set compass to a certain wide so that the resulting arch intersects with the side AB in D and with the side AC in E
At point D, set compass to the similar wide with that of procedure 2 to produce an arch.
At point E, set compass to the similar wide with that of procedure 3 to produce another arch. Call the intersection point of the arches as M.
Connect point A to point M.
Line segment AM intersects with line segment BC at point T.
Line segment AT is called the bisector of triangle ABC at A.

Median and Perpendicular Bisector of Triangles

Perpendicular bisector of triangle is a line passing through the mid-point and perpendicular to a certain side of a given triangle. A median of a triangle is a line passing through one vertex of the triangle and the midpoint of the side opposite this angle.

[Gambar garis sumbu dan garis berat]

In order to construct the median CD you need first to construct perpendicular bisector PQ to indicate the mid-point of AB, i.e. D. After you can indicate point D, you then are to construct the median CD.

The stages of constructing perpendicular bisector and median of a triangle:
Construct a triangle ABC
At point A, set compass to a certain wide so that the resulting arch intersects with the side AB
At point B, set compass to the similar wide with that of procedure 2 to produce another arch so that it intersects with the first arch in P and Q.
Connect point P to point Q
Line segment PQ is perpendicular bisector of line segment AB
Call it D for the intersection between PQ and AB
Connect point D to point C
Line segment CD is the median of triangle ABC passing through C.

Exercise

Construct all bisectors, perpendicular bisectors and medians of the following triangles!

[Gambar 3 biah segitiga]

Perimeter and Area of Triangles

Determining the Perimeter of Triangles

Perimeter of a triangle is the sum of its three sides.

[Gambar keliling segitiga]

Perimeter (P) of triangle ABC is formulated as P = AB + AC + BC

Example
Determine the perimeter of the following triangle!
[Gambar segitiga]
P = AB + AC + BC
= 10 + 4 + 8
= 22

Hence, the perimeter of ABC is 22 cm

The perimeter of a triangle-like garden is 60 m. The length of the two sides of the garden are 15 m and 28 m. Determine the length of its other side!
Solution:
Let, the length of the unknown side is b, then
P = b + 15 + 28
60 = b + 15 + 28
= b + 43
b = 60-43 = 17
Hence, the length of the other side of the garden is 17 m.
Exercise
Find out the perimeter of the following triangles!
16 cm, 10 cm, and 20 cm
30 cm, 45 cm, and 35 cm
75 cm, 100 cm, and 120 cm
The perimeter of a triangle is 55 cm. The length of the two sides are 14 cm and 16 cm. Determine the length of the third side!

Determining the Area of Triangles

At any triangle there are base and height. Each side of triangle can be its base. The height of triangle is the straight line perpendicular to the base, passing through the opposite apex.
Followings are the examples of triangles with their base and height.

[Gambar tiga segitiga dengan alas dan garis tinggi]

You can determine the area of triangle using the following formula

Area of triangle = ½ x base x height

Example
Determine the area of triangle in which its base is 8 cm and the height is 6 cm!
Solution:
[Gambar segitiga yang dimaksud]
Area of triangle = ½ x base x height = ½ x 8 x 6 = 24

Hence, the area of triangle is 24 cm2

Exercise
Determine the area of the following triangles!
[Gambar segitiga 1 sd 5]

Angles of Triangles

The Sum of the Angles in a Triangle

The sum of the three angles of a triangle is 180°. Consider the following figures:

[Gambar guntingan segitiga]

Tear off each angle, and then put the three angles together like the following figure.

You noticed that the sum of the three angles is 180°.
Example
Determine the measure of angles in the following triangle!
[Gambar segitiga siku-siku]
Solution:
Triangle ABC is the right triangle at A. Therefore, ∠ CAB=90°
On the other hand we have
∠ CAB+∠ABC+ ∠BCA=180° then
90° + 2 x° + x° = 180°
90° + 3 x° = 180°
3 x° = 90°
x° = 30°
Hence the measure of each angle of triangle ABC are:
∠ CAB=90° , ∠ BCA=x°=30° , and ∠ ABC=2x°=2(30°)=60°

The comparison among angles of a triangles is 2 : 3 : 4
Determine the measure of each angle!
Solution:
Let, the measure of the angles are 2x°, 3x° and 4x°.
On the other hand we have the sum of the three angles of a triangle is 180°, then
2x°+ 3x°+ 4x° = 180°
9x° = 180°
x° = (180°)/9=20°

Hence, 2x°=2(20°)=40°; 3x°=3 (20°)=60°; 4x°=4(20°)= 80°

Thus, the measure of the angles are 40°, 60° and 80°.


Exercise
Determine the value of y in the following triangles!
[Gambar Segitiga a, b c]

Then, determine also the measure of each angle!
Proportion of angles is 5: 2: 3
Determine the angles.

Interior and Exterior Angles of Triangles

In the following triangle, ∠ BAC,∠ BCA,and ∠ ABC are called interior angles of triangle ABC, while ∠ ACD is called exterior angle of triangle ABC.

[Gambar segitiga]

An exterior angle is equal to the sum of the two interior angles that do not share a side
with the exterior angle.

You know that the sum of the three angles of a triangle is 180°.
or , ∠ CAB+ ∠ ACB+ ∠ ABC= 180°
Due to ∠ ACB shares a side with ∠ ACD, then ∠ ACB+ ∠ ACD= 180°
Hence, ∠ ACD= ∠ BAC+ ∠ ABC

Example
Consider the following figure

[Gambar segitiga]

Determine the measure of ∠ ACD and the measure of ∠ ACB,
when ∠ ABC= 80° and ∠ CAB= 35°!
Solution:
The sum of the three angles of a triangle is 180°, therefore,
∠ CAB+ ∠ ACB+ ∠ ABC= 180°
= 35°+80°=115°
Due to ∠ ACD= ∠ CAB+∠ ACB, hence
∠ ACB= 180°- ∠ ACD
= 180°-115°=65°

Exercise

Triangle ABC, ∠ ABC= 75° and ∠ ACB= 55°. Determine:
∠ BAC;
Exterior angle of A;.
Exterior angle of B.

Determine the exterior angles of P in the following triangles!

[Gambar segitiga]

Quadrilateral

Rectangle

Rectangle is a quadrilateral with two pairs of parallel sides, four right angles, opposite sides equal in length, equal diagonals bisecting one another. Consider the following rectangle ABCD.

[Gambar persegi panjang]

The elements of rectangle consist of length, width, and diagonal.
AB and CD of the rectangle ABCD are called the length
AD and BC of the rectangle ABD are called the width
AC and BD of the rectangle ABD are called diagonal

Followings are the properties of rectangle.
Opposite sides of rectangle are equal and parallel
Each interior angle of rectangle is a right angle
Diagonals of rectangle are equal
Diagonals of rectangle have the same midpoint
Let, a rectangle has its length p and width l then:
Perimeter (K) of the rectangle is K = 2 (p + l)
Area (L) of the rectangle is L = p x l
Example
Draw a rectangle ABCD in which AD = 4 cm and CD = 6 cm!
Is the length of AB equal to CD?
Is the length of AD equal to BC?
Determine the perimeter and the area of the rectangle ABCD!

Solution:
The following is a drawing of rectangle ABCD

[Gambar persegi panjang ABCD]

The length of AB is equal to the length of CD because the opposite sides of rectangle are equal
The length of AD is equal to the length of BC because the opposite sides of rectangle are equal
The length of rectangle ABCD is 6 cm. The width of rectangle ABCD is 4 cm. Therefore,
we can determine the perimeter and the area as:
K = 2 (p + l)
= 2 (6 + 4)
= 2 (10)
= 20
L = p x l
= 6 x 4
= 24
Hence, the perimeter and the area of rectangle are 20 cm and 24 cm2

Exercise
Complete the following table of rectangle!

[Tabel ]

Mr Karto has a field in the form of rectangle. The length of the field is twice as the width.
If the perimeter of Mr. Karto’s fiels is 48 m, determine:
the length and the width of the field;
the area of the field;
if the field of Mr. Karto produces 5 kg casava for each m2
, how many kg casava will produced by Mr. Karto’s field totally?
Square

Square is two-dimensional figure with four straight sides, whose four interior angles are right angles (90°), and whose four sides are of equal length. Consider the following square ABCD.

[Gambar persegi]

The elements of square are:
AB, BC, CD, and AD are the sides of square ABCD.
AC and BD are diagonals of square ABCD.

The properties of square:
all four sides of a square are of equal length
the diagonals of a square are the bisectors
the diagonals of a square are perpendicular bisector of each other.

Let, a square has its side of s cm length:
The perimeter of the square K = 4 s
The area of the square L = s2

Example

The side of a square is 16 cm. Then, each angle is torn off 2 cm. Determine the perimeter and the area of the torn off square!

Solution
The following figure is resulted from the torn off square of 16 cm length of side.

[Gambar bangun persegi yang telah terpotong]

You can indicate the area of the last figure by subtracting the area of the square before it is torn off by the total area of torn off parts.
The area of square before it is torn off is 16 cm x 16 cm = 256 cm2
The area of each torn off part is 2 cm x 2 cm = 4 cm2
Therefore, the total area of torn off parts = 4 x 4 cm2 = 16 cm2
The area of figure after it is torn off = 256 – 16 = 240 cm2
The perimeter of the figure after it is torn off is:
K = 4 x (2 + 12 + 2)
= 4 x 16
= 64 cm
Exercise
Complete the following table!

[ Tabel soal}

A bricklayer is to set the square marbles of the size 20 cm x 20 cm each, to fit together without leaving any space of a rectangle-flour with the size of 4 m x 3 m.
Calculate the number of marbles to tessellate the flour.
Determine the area of the flour to be tessellated.
[Gambar lantai]
Parallelogram
We can form the parallelogram by combining two congruence triangles.
The followings are the elements of parallelogram.
AB, BC, CD and AD are the sides of parallelogram
AC and BD are the diagonals of parallelogram
AB is the base of parallelogram
t is the height of parallelogram

Note:
Triangles which have the exact same size and shape are congruence

[Gambar jajaran genjang]

The properties of parallelogram are:
The opposite sides are equal and parallel to each other
The opposite angles are equal
The sum of the consecutive angles is 1800
The diagonals are perpendicular bisector to each other

We can determine the perimeter of parallelogram by adding up all sides.

The perimeter of parallelogram, K = 2 (AB + BC)

[Gambar jajaran genjang]

The area of parallelogram can be found by the following formula.

The area of parallelogram, L = base x height
= a x t

Example
The length of AB and AD of a parallelogram ABCD are consecutively 12 cm and 7 cm. The height of parallelogram is 5 cm. Determine the perimeter and the area of parallelogram ABCD!
Solution:
You can draw the parallelogram ABCD as follow.

[Gambar jajaran genjang]



Hence, the perimeter of the parallelogram is 38 cm and the area is 60 cm2

Exercise
Complete the following table of parallelogram


[Tabel jajarangenjang]

Let, PQRS is a parallelogram in which its diagonals intersect at point O.
Indicate two couples of parallel sides of the parallelogram PQRS!
Indicate two couples of obtuse triangle in the parallelogram PQRS!
Indicate two couples of equal angles of the parallelogram PQRS!
Consider the following parallelogram

[Gambar jajarangenjang]

Calculate the value of x.
And then, calculate the length of AB.
Calculate the area of the parallelogram.

Trapezoid

Trapezoid is quadrilateral figure with two parallel sides. or bases, of unequal length.

The elements of trapezoid are lower base, upper base, height and legs. Consider the following trapezoid figure.

[Gambar Trapesium]


AB, BC, CD, and AD are the sides of trapezoid
AB has a special name, i.e. the lower base of trapezoid
CD has a special name, i.e. the upper base of trapezoid
AD and BC have a special name, i.e. the legs of trapezoid
Line t is called the height of trapezoid

Type of Trapezoid Can be formed from The Figure

Right Trapezoid A rectangle and a right triangle
gambar

A square and a right triangle
gambar

Isosceles Trapezoid A rectangle and two congruence right triangles
gambar
A square and two congruence right triangles
gambar
Scalene Trapezoid Some plane figures
gambar

The properties of Trapezoid

[Gambar trapesium]

Right Trapezoid
It has exactly two right angles, i.e. ∠ BAD and ∠ ADC
∠ BAD+∠ ADC=1800
∠ ABC+∠ BCD= 1800


[Gambar trapesium]

Isosceles Trapezoid
∠ BAD= ∠ ABC
∠ ADC= ∠ BCD
∠ BAD+ ∠ ADC=1800
∠ ABC+ ∠ BCD=1800
The two diagonals are equal in length (AC = BD)

[Gambar trapesium]

Scalene Trapezoid
∠ BAD+∠ ADC=1800
∠ ABC+ ∠ BCD=1800

The perimeter and area of trapezoid are:
K = AB + BC + CD + AD
L = ½ x (AB + DC) x t

[Gambar trapasium]

Example
Consider the following figure

[Gambar trapesium]

Determine:
The perimeter of trapezoid ABCD
The area of trapezoid ABCD
Solution:
The perimeter of trapezoid can be determined by adding up all sides

K = AB + BC + CD + AD
= 8 + 5 + 5 + 4
= 22
Hence, the perimeter of trapezoid ABCD is 22 cm

Trapezoid ABCD is the right trapezoid. Therefore, the height of trapezoid is (t) equals AD.

L = ½ x (AB + CD) x t
= ½ x ( 8 + 5)) x 4
= ½ x 13 x 4
= 26
Hence, the area of trapezoid ABCD is 26 cm2


BAGIAN KE TIGA (Hal 160 sd 166)
Bagian Terakhir (Hal 167 -172: Soal-soal) menyusul
Exercise

Complete the angles of the following trapezoid

[Gambar 3 bua trapesium]

Trapezoid PQRS is isosceles trapezoid in which PQ // RS. The length of PQ is 15 cm, PR = 1/3 PQ, and the height of trapezoid PQRS is 12 cm. Determine the area of trapezoid PQRS!
The field of Mr. Arman has its square form with 30 m length of side. Mr. Arman then buys new field to extend the field on the left and on the right hand such that the ultimate form of Mr. Aman’s field is isosceles trapezoid as it is shown in the figure. Mr. Arman wish to fence the field, calculate how long the fence that Mr. Aman needs?

Kite

A kite can be formed from two congruence triangles by joining them through their bases.

[Gambar layang-layang]

The elements of a kite:
PQ, PS, QR and SR are called the sides of the kite
PR and QS are called diagonals of the kite

The properties of a kite are:
Two pairs of adjacent sides are equal
One of the diagonals is a perpendicular bisector of the other.
Two pairs of opposite angles are equal

The perimeter and the area of a kite are:

K = 2 (PQ + QR)
L = ½ x QS x PR

Note:
Axis of symmetry is a line producing a figure identical to the original or a mirror image of the original figure.

Example
Consider the following figure.

[Gambar layang-layang]

Determine:
the perimeter of the kite PQRS
the area of the kite PQRS
Solution:
K = 2(PQ + QR)
= 2(13 + 20)
= 2 (33)
= 66
Hence, the perimeter of the kite PQRS is 66 cm

L = ½ x QS x PR
= ½ x (12 + 12) x (5 + 16)
= ½ x 24 x 21
= 252
Hence, the area of the kite PQRS is 252 cm2

Exercise
Consider the following figure

[Gambar layang-layang]

Determine the unknown angles of the kite ABCD
Consider the following figure

[Gambar layang-layang]
Determine:
the perimeter of the kite PQRS
the area of the kite PQRS

Rhombus
Rhombus can be formed by combining two congruence isosceles triangles through its bases.
[Gambar belah ketupat]
The elements of rhombus:
AB, BC, CD, and AD are the sides of rhombus ABCD
AC and BD are the diagonals of rhombus ABCD
The properties of rhombus:
The four sides of rhombus are equal
The opposite angles of rhombus are equal
The diagonals are perpendicular bisector to each other

The perimeter and the area of rhombus are:

K = 4 s
L = ½ x AC x BD
Example
Consider the following figure
[Gambar belah ketuat]
Determine:
the perimeter of rhombus ABCD
the area of rhombus ABCD
Solution:
The side of rhombus ABCD is 15 cm
K = 4 s = 4 x 15 = 60
Hence, the perimeter of rhombus ABCD is 60 cm
We know that ½ AC = 12 cm and ½ BD = 9 cm, hence AC = 2 x 12 cm = 24 cm
and BD = 2 x 9 = 18 cm.
L = ½ x BD x AC
= ½ x 18 x 24
= 216
Hence, the area of rhombus ABCD is 216 cm2

Exercise

Complete the following table of rhombus

[Tabel belah ketupat]

A cassette-shelf is in the form of rhombus as it is shown by the following figure.
[Gambar rak kaset]
Determine the perimeter and the area of that cassette-shelf!
(Hint: to determine the length of OD, use the formula OD = ...)

EXERCISE 9
Chose the correct answer

The measure of the angles of a triangle are consecutively 2xo, (x + 40)o and (4x + 35)o.
The value of x is ...
55
40
35
15

Of the isosceles triangle ABC, the length AB = BC and ∠ ABC=30o. The measure of ∠ BAC is…
150o
120o
75o
60o

Of the isosceles triangle PQR, the length PR = QR and ∠ ABC=37,5o. The measure of
∠ PRQ is…
a.
b.
c.
d.
(lihat dok)

The perimeter of triangle ABC is 120 cm. If the proportion of the sides AB : BC : AC = 3: 4 : 5, the length of AB is...
a.
b.
c.
d.
(lihat dok)

The following ∆ABC can be constructed, except..
a.
b.
c.
d.
(lihat dok)

Consider the following figure
[Gambar segitiga]

The area of ∆KLM is ...
a.
b.
c.
d.
(lihat dok)

The area of a triangle is 135 cm2. The base of the triangle is 18 cm2. The height of the triangle is...
a.
b.
c.
d.
(lihat dok)

The area of ∆ABC in the following figure is..
a.
b.
c.
d.
(lihat dok)

The area of ∆ABC in the following figure is..
a.
b.
c.
d.
(lihat dok)

The length and the width of a rectangle are consecutively (3x + 4) cm and (2x + 4) cm. The perimeter of the rectangle is 36 cm. The value of x is ...
a.
b.
c.
d.
(lihat dok)

A yard has its form of rectangle with the size of 10 m x 8 m. In the middle of the yard is built a pond with the size of 4m x 6m. Therefore, the area of the yard out of the pond is..
a.
b.
c.
d.
(lihat dok)




The proportion of the length and the width of a rectangle is 2:5. If the area of the rectangle is 40 cm2, hence the width of the rectangle is..
a.
b.
c.
d.
(lihat dok)
A square yard has it sides to be planting with akasias tree. The distance between every two akasias is 3 m. If the side of the yard is 15 m, then the number of akasia tree is..
a.
b.
c.
d.
(lihat dok)
The length and the width of a rectangle are 30 cm and 18 cm. If you have a square PQRS in which its perimeter is ½ of the perimeter of the rectangle ABCD, then the side of the square PQRS is..
a.
b.
c.
d.
(lihat dok)
The perimeter of shaded-area in the following figure is...
[Gambar]
a.
b.
c.
d.
(lihat dok)
Consider the following figure.
[Gambar trapesium]
The area of trapezoid showing in the figure is...
a.
b.
c.
d.
(lihat dok)
The area of the following shaded-area is ...
a.
b.
c.
d.
(lihat dok)
Consider the following figure of a kite
[Gambar layang-layang]
If the perimeter of the kite is 240 cm and AD = ½ BC then the value of ½ x + 23 is...
a.
b.
c.
d.
(lihat dok)
[Gambar trapesium]
The area of shaded-area in the above figure is...
a.
b.
c.
d.
(lihat dok)

The perimeter of a rhombus is 52 cm. If the length of its diagonal AC = 10 cm then the area of rhombus ABCD is..
a.
b.
c.
d.
(lihat dok)

Solve the following problems correctly

The perimeter of ∆ABC is 35 cm. The length AB = BC. Determine the length of BC if the length of AC is 10 cm!
The perimeter of a rectangle is 62 cm. The width of the rectangle is 12 cm. Determine:
the length of the rectangle
the area of the rectangle
The length of one of the diagonals of rhombus is 10 cm. While the proportion of the length of both diagonals is 2 : 5. Determine the area of the rhombus!
Consider the following figure of a kite!

[Gambar layang-layang]

Determine the area of shaded-area of the kite if the length of Ac is 10 cm, the length of BD is 4 cm, and the proportion of AO and OC is 2:3!

The size of the flour if 12 m x 9 m. The flour will be tessellated using the marbles with the size of 30 cm x 30 cm each. Determine the number of marbles to be used to fit together without leaving any space of the flour.