AESTHETIC EDUCATION9th GRADE: Q1 / P1

TECHNICAL DRAWING:Technical drawing is a drawing or plan that is used to communicate direction and specifics to a group of people who are creating something, to explain how something works or how to build something.

Who Uses Technical Drawings Engineers Contractors Plumbers Electricians Landscape architects Inventors

An example of a technical drawing is a drawing made for a plumber with unique symbols to show where all the water lines, sinks, faucets, tubs and toilets are to be located. An example of a technical drawing is a drawing made with computer-assisted design (CAD) to show the details of a new home building project.

INSTRUMENTS:Drawing Board:

Drawing board is made from strips of well-seasoned soft wood generally 25 mm thick. One of the shorter edges of the rectangular board is provided with perfectly straight ebony edge which is used as working edge on which the T-square is moved while making Drawings.

T-square:

T-squares are made from hard wood. A T-square consists of two parts namely the stock and the

blade joined together at right angles to each other. The stock is made to slide along the working edge and the Blade moves on the Drawing board.

Set Squares:

Set squares are generally made from Plastic or celluloid material. They are triangular in shape with one corner, a right angle triangle. A pair of set squares (30°–60°) and 45°.

They are used to draw lines at 30°, 60° and 45° to the vertical and horizontal.

Protractor:

Protractors are used to mark or measure angles between 0 and 180°. They are semicircular in shape (of diameter 100mm) and are made of Plastic or celluloid . Protractors with circular shape capable of marking and measuring 0 to 360°

Drawing Pencils:

The accuracy and appearance of a Drawing depends on the quality of Pencil used to make Drawing.

The grade of a Pencil lead is marked on the Pencil.

HB denotes medium grade. Increase in hardness is shown by value put in front of H such as 2H, 3H etc.

Softer pencils are marked as B, 2B, and 4B etc.

Pencil marked 3B is softer than 2B and Pencil marked 4B is softer than 3B and so on.

Beginning of a Drawing may be made with H or 2H. For lettering and dimensioning, H and HB Pencils are used.

Drawing Pins and clips:

These are used to fix the Drawing sheet on the Drawing board.

Compass:

Compass is used for drawing circles and arcs of circles. The compass has two legs hinged at one end. One of the legs has a pointed needle fitted at the lower end whereas the other end has provision for inserting pencil lead.

Fineliner:

Fineliner pens are a class of fine fiber or plastic tip pens that are typically used for graphic, drawing or sketching purposes, but are also popular for handwriting as many people like the unique feel of the tip compared to a traditional ball-tipped pen.

The tips are generally long and metal-clad to allow use with rulers and templates without bending. Fineliners are generally relatively cheap, as the construction is very simple. They're mostly disposable, but there are a few premium refillable options. Most fineliner pens use dye-based ink, which is not hugely permanent, but there are many that use lightfast and waterproof pigment ink, especially the more technical drawing ranges.

AESTHETIC EDUCATION9th GRADE: Q1 / P1

TYPE OF LINES

A line is a series of points adjacent to each other. Where a point has no dimension, a line has one dimension. They have a length, but nothing else. In reality a line would need a second dimension to actually see it, but we’ll continue to call them lines and not something else here.

Lines are used to draw but they can also fill in spaces in a drawing and add texture.

We are going to make a small classification for the lines depending on a characteristic.

LINES BY THE POSITION IN THE SPACE:

LINES BY THE SHAPE THEY HAVE:

Note: there are more types of line by shape this is only a small list

LINES BY THEIR EXTENSION:

LINES BY THEIR RELATION WITH EACH OTHER:

WHAT IS AN ANGLE?

An angle is the amount of turn between two straight lines that have a common end point (the vertex).

AOB = is used to represent an angle.

A & B are the lines and O is the vertex.

TYPES OF ANGLES

Zero Angle: is the angle that measures 0º

AOB = 0º

Acute Angle: the measure of the angle is less than 90 °

AOB < 90º

Right Angle: is the angle that measures exactly 90 °

AOB = 90º

Obtuse Angle: the measure of the angle is greater than 90 °

AOB > 90º

Straight Angle: It is the angle that measures 180º

AOB = 180º

Reflex Angle: is one which is more than 180° but less than 360°

AOB > 180º AOB < 360º

Full Angle: It means turning around until you point in the same direction again or 360º

AOB = 360º

AESTHETIC EDUCATION

9th GRADE: Q1 / P1

A vertex is a corner. An edge is a line segment that joins two

vertices. A face is an individual surface.

TYPE OF TRIANGLES

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C

A triangle has three sides and three angles The three angles always add to 180°

There are three special names given to triangles that tell how many sides (or angles) are equal.

There can be 3, 2 or no equal sides/angles:

Triangles can also have names that tell you what type of angle is inside:

AESTHETIC EDUCATION9TH GRADE: Q1P1

CLASSIFICATION OF ANGLES BY THE POSITION THEY KEEP BETWEEN EACH OTHER.

NOTE: Congruent AnglesCongruent Angles have the same angle (in degrees or radians). That is all.

1. Complementary Angles.

Two angles are Complementary when they add up to 90 degrees (a Right Angle).These two angles (40° and 50°) are Complementary Angles, because they add up to 90°:

Notice that together they make a right angle.But the angles don't have to be together.These two are complementary because 27° + 63° = 90°

2. Supplementary Angles

Two Angles are Supplementary when they add up to 180 degrees.

These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°:

Notice that together they make a straight angle.But the angles don't have to be together.

3. Corresponding Angles

When two parallels lines are crossed by another line (called the Transversal):The angles in matching corners are called Corresponding Angles and are equal.

a = e b = f c = g d = h

4. Alternate Interior Angles

When parallels lines are crossed by another line (called the Transversal):The pairs of angles on opposite sides of the transversal but inside the parallels lines are called Alternate Interior Angles and are equal.

c = f d = e

5. Alternate Exterior Angles.

When parallel lines are crossed by another line (called the Transversal):The pairs of angles on opposite sides of the transversal but outside the parallel lines are called Alternate Exterior Angles and are equal.

a = h b = g

6. Vertically Opposite Angles

Vertically Opposite Angles are the angles opposite each other when two lines cross.

NOTE: "Vertical" in this case means they share the same Vertex (or corner point), not the usual meaning of up-down.

In this example, a° and b° are vertically opposite angles.

The interesting thing here is that vertically opposite angles are equal:

a° = b° c° = d°(in fact they are congruent angles)

AESTHETIC EDUCATION9TH GRADE: Q1P2Materials: A4 white cardboard, 2h pencil, compass, ruler, set squares.

PERPENDICULARS

Lines that are at right angles (90°) to each other.

1. Perpendicular passing through the middle of the segment AB.

a) Draw the segment AB

b) With the compass center at A and with a larger opening than half of the segment AB draw two arcs, above and below the segment AB.

c) With the same opening on the compass center at B, cut the previous arcs to obtain the points 1 and 2

d) Join the points 1 and 2, and we find the perpendicular.

NOTE: This procedure is also called perpendicular bisector

Definition: A line which cuts a line segment into two equal parts at 90°.

2. Perpendicular passing through a point c, on the AB segment

a) Draw the segment AB. Place the point C on the segment AB.

b) With the compass make center in C and with any opening cut with two arcs the segment AB and we place points 1 and 2.

c) Center in point 1, with an opening greater that the distance between 1 and 2, draw an arc above the segment AB

d) With the same opening repeat the same process now from point 2 and cut the previous arc and locate point 3.

e) Connect with a straight line the point C and 3, and find the perpendicular.

3. Perpendicular passing through a point C, outside the AB segment.

a) Draw AB segment and place the C point anywhere outside the AB segment.

b) Center at C, draw an arc that cut the AB segment.

c) Mark point 1 and 2 at the intersections of AB segment and the arc drawn.

d) Center at 1, and with any opening draw an arc below AB segment. Repeat from point 2 and mark point 3 at the crossing of the arcs.

e) Draw a line from C to point 3.

4. Perpendicular passing at the endpoint of AB segment.

a) Draw AB segment

b) Center at B, draw a semicircumference that cross AB, mark point 1.

c) With the same opening center at 1, and cut the semicircumference, mark 2, then repeat from 2 and find point 3.

d) With the same opening and center at 2, draw and arc above the semicircumference, repeat from 3 and cross the two arcs, mark point 4.

e) Draw a line from B to point 4.

5. Perpendicular passing at the endpoint of AB segment using a C point outside the segment.

a) Draw AB segment, place the point C anywhere outside AB near the endpoint of the segment.

b) Center at C and draw a semicircumference that pass through B, mark point 1.

c) Draw a line from 1 to C and extend until it cut the semicircumference, mark point 2

d) Draw a line from B to point 2

AESTHETIC EDUCATION9TH GRADE: Q1P2Materials: A4 white cardboard, 2h pencil, compass, ruler, set squares.

PARALLELS

Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. (They also point in the same direction). Just remember:

Always the same distance apart and never touching.

1. Parallel passing through a point c, outside the AB segment.

a) Draw AB segment and place point C anywhere outside AB.

b) Center at C and draw a large arc which cuts AB segment, place point 1.

c) With the same opening center at 1 and draw a large arc that cut AB and pass through C, place point 2.

d) Center at 2 and set the width of the compass to the distance from 2 to C.

e) Center at 1 and draw an arc that cut the large arc, mark point 3.

f) Draw a line from 3 to C.

2. Parallel from a C point place on the AB segment.

a) Draw AB segment and place point C.

b) Center at C and with any opening draw a semicircumference that cut AB, place point 1 & 2.

c) Center at 1 and with an opening from 1 to 2 draw an arc, then repeat from 2 and cut the arc, place point 3 and 4.

d) Draw a line from 3 to 4 and extend.

3. Parallel from AB segment with a known X distance and 2 points on it.

a) Draw AB segment and place point 1 and 2 on the segment.

b) Draw a line segment with X distance of 2.8 cm

c) Set the width of the compass to the X distance, place the metal point on one end and the graphite on the other end of the X segment.

d) Center at 1 and draw an arc above AB segment, repeat from point 2.

e) Draw a line that pass on top the two arcs drawn.

4. Parallel from a given line that pass through a given point.

a) Draw AB segment, place point C on the segment and a point D outside it.

b) Draw a line from C to D and extend it.

c) Center at C and with any opening draw an arc which cuts AB segment and the given line CD, mark point 1 and 2.

d) Retain the width of the compass, center at D and draw an arc a similar arc that cut CD, place point 3.

e) Set the width of the compass to the distance from 1 to 2.

f) Center at 3 and draw an arc that cut the previously drawn arc, place point 4.

g) Draw a long line that passes by points D and 4.

AESTHETIC EDUCATION9TH GRADE: Q1P3Materials: A4 white cardboard, 2h pencil, compass, ruler, set squares.

JUST TO RECALL

WHAT IS A TRIANGLE?

A triangle is one of the basic shapes of geometry. It is a 3-sided polygon. Every triangle has 3 sides and 3 angles. The 3 angles always add to 180°.

TYPES OF TRIANGLES:

Equilateral, Isosceles and Scalene:

We distinguish three special names given to triangles that tell how many sides (or angles) are equal.

TYPES OF TRIANGLES: By sides

Equilateral Triangle:

3 equal sides and 3 equal angles (always 60°)

Isosceles Triangle:

2 equal sides and 2 equal angles

Scalene Triangle:

No equal sides and no equal angles

TYPES OF TRIANGLES: By Angles

Acute, Right and Obtuse:

Triangles can also have names that tell you what type of angle is inside.

Acute Triangle:

All angles are less than 90°

Right Triangle:

Has a right angle (90°)

Obtuse Triangle:

Has an angle more than 90°

IMPORTANT LINES OF A TRIANGLE

GOALS Identify the altitudes, medians, perpendicular bisectors and angle bisectors in a triangle.

Short overview on this topic:

AltitudeThe perpendicular segment from a vertex to its opposite side

MedianA median joins a vertex and the opposite side at his midpoint.

Perpendicular BisectorA perpendicular bisector remains at right angles to a side and divides it into two equal halves.

Angle BisectorAn angle bisector divides an angle into two equal halves.

ALTITUDE

Any side of triangle can be assumed as its base. Since, a triangle has three sides, hence it can have three bases. The altitude of a triangle is defined as a perpendicular drawn from any vertex (a point where two sides of a triangle meet) on to the opposite side (base) of that triangle, i.e. an altitude may be referred as a line segment which passes through any vertex and forms the right angle with the edge opposite to this vertex.

Let us have a look at following diagram in order to understand the concept of altitude more clearly.

The altitude may fall outside the triangle too. In some cases, such as obtuse-angled triangle, the altitude does not directly meet in the base.

For this purpose, the base has to be extended as shown in the following figure:

The intersection point of altitude and the base (or the extended base) of the altitude is termed as the foot of the altitude. The distance between the vertex and the foot of altitude is eventually the length of altitude. There are three altitudes in a triangle. The biggest altitude of a triangle must be perpendicular to its smaller side. Most usually, the altitude of triangle is denoted by letter h, since it represents the height of the triangle.

There are some basic properties of altitudes. They are as follows:

Altitudes can be used to compute the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area.

For acute triangles the feet of the altitudes all fall on the triangle's interior.

For right triangles the feet of the altitudes fall on the triangle's edge.

In an obtuse triangle (one with an obtuse angle), the foot of the altitude to the obtuse-angled vertex falls on the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side, exterior to the triangle.

The three Altitudes of a triangle are concurrent. i.e. they meet at a single point known as the Orthocenter.

MEDIAN

Median of a triangle is a line segment joining a vertex of a triangle to the midpoint of the opposite side. A triangle has three sides, so every triangle has exactly three medians, each running from one vertex to the side exactly opposite.

Note: In the case of equilateral and isosceles triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.

In the triangle ABC, D, E and F are the mid-points of sides BC, CA and AB respectively. The line segments AD, BE and CF are the medians of the triangle.

There are some basic properties of medians which make them very important. They are as follows:

Each median divides the triangle into two parts, and the area of these two parts are exactly equal.

In this triangle ABC, the median CM divides the triangle into two parts: △ACM and △BCM. These two triangles, △ ACM and △BCM have equal area.

The three medians of a triangle are concurrent. i.e. they meet at a single point known as the Centroid .

The medians AL, BM and CN meet at P, the centroid of the triangle. This can be proved by the corollary of Ceva's Theorem.

The centroid is the center of gravity of the triangle.

In the above figure, G is the centroid as well as the center of gravity.

PERPENDICULAR BISECTOR

In general, a perpendicular bisector is defined as a line segment that bisects as well as perpendicular to another line segment. In case of triangles too, the definition remains the same. Since a triangle is made up of three line segments, eventually a triangle would have three perpendicular bisectors. They are demonstrated in the diagram below:

The perpendicular bisectors of a triangle have following important properties.

It divides the side into two equal parts and it is inclined at 90º to the side.

The perpendicular bisectors of a triangle are concurrent, i.e. meet at one point. This point of intersection is called circumcenter.

Circumcenter is located at an equal distance from all three vertices.

In case of an acute triangle, the circumcenter lies inside the triangle. If the triangle is obtuse, then it lies outside the triangle. Also, circumcenter lies right in the middle of hypotenuse when the triangle is right angled triangle.

If we draw a circle assuming a distance of any vertex from circumcenter as radius and circumcenter as a center, then we obtain a circle passing through all three vertices outside the triangle. This circle is known as circumcircle.

ANGLE BISECTORS

A bisector is nothing but a ray or line that is used to partition an object into two equal parts. A triangle has a total of three bisectors. The bisectors of the triangle bisects the interior angles of the triangle. Since there are three angles in a triangle, the number of bisectors in a triangle is also three.

In the above figure, the bisector of the triangle ABC is AD. This is because, the ray AD bisects the angle L A in to two equal parts.

Angle bisector of a triangle should be the line that bisects the inner angles.

These bisectors should be intersect at center of the circle formed inside the triangle.

The center of the circle is called as incircle and is denoted by S in the figure.

Every side of triangle should be cut by the corresponding angle bisector into two segments.

CONCLUDING OBSERVATIONS OF THE TOPIC:

In Geometry, the altitudes of a triangle are the straight lines through a vertex and which is perpendicular to (i.e. forming a right angle with) the opposite side or an extension of the opposite side. In other words

it starts at a vertex and is perpendicular to the opposite side

Furthermore, the three altitudes intersect in a single point, called the orthocenter of the triangle.

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In Geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. These medians intersect in a single point, called the centroid.

In Physics, the centroid of a triangle is the point through which all the mass of a triangular plate seems to act. It is also known as the 'center of gravity', 'center of mass', or barycenter.

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A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint. The three perpendicular bisectors meet in a single point, called the triangle's circumcenter. The circumcenter of a triangle is the point of concurrency of the perpendicular bisectors of the sides of a triangle. This point is the center of the circumcircle, i.e. the circle passing through all three vertices.

In general, 'to bisect' something means to cut it into two equal parts. The 'bisector' is the thing doing the cutting. With a perpendicular bisector, the bisector always crosses the line segment at right angles (90°).

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An angle bisector divides the angle into two angles with equal measures. The angle bisectors of the angles of a triangle are concurrent in one single point called the incenter of the triangle, which is the center of the incircle.

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle, i.e. it touches (is tangen to) the three sides.

TRIANGLE CONCURRENCY POINTS or THE FOUR TRIANGLE CENTERS

These are the four concurrency points in a triangle.

ORTHOCENTERThe orthocenter is the point at which the three altitudes of a triangle intersect.

The orthocenter is the center of the triangle created from finding the altitudes of each side. Usually denoted by H. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side. An altitude of the triangle is sometimes called the height. Remember, the altitudes of a triangle do not go through the midpoints of the sides unless you have a special triangle, like an equilateral triangle.

The orthocenter does not have to be inside the triangle. Check out the cases of the obtuse and right triangles below. In the obtuse triangle, the orthocenter falls outside the triangle. In a right triangle, the orthocenter falls on a vertex of the triangle.

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CENTROIDThe centroid is the point at which the medians of a triangle intersect. It is the center of gravity of the triangle.

The centroid of a triangle is constructed by taking any given triangle and connecting the midpoints of each side of the triangle to the opposite vertex. Usually denoted by G. The line segment created by connecting these points is called the median. You see the three medians as the dashed lines in the figure above.

No matter what shape your triangle is, the centroid will always be inside the triangle. You can look at the above example of an acute triangle, or the below examples of an obtuse

triangle and a right triangle to see that this is the case.

The centroid is the center of a triangle that can be thought of as the center of mass. It is the balancing point to use if you want to balance a triangle on the tip of a pencil, for example.

The centroid is always inside the triangle.

Each median divides the triangle into two smaller triangles which have the same area.

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CIRCUMCENTERThe circumcenter is the point at which the three perpendicular bisectors of the sides of a triangle intersect. Usually denoted by O. It is also the center of the circle circumscribed around a triangle. This circle passes through all three vertices of the triangle

The circumcenter is the center of the circle such that all three vertices of the circle are the same distance away from the circumcenter. Thus, the circumcenter is the point that forms the origin of a circle in which all three vertices of the triangle lie on the circle. Thus, the radius of the circle is the distance between the circumcenter and any of the triangle's three vertices. It is found by finding the midpoint of each side of the triangle and constructing a line perpendicular to that side at its midpoint. Where all three lines intersect is the circumcenter.

The circumcenter is not always inside the triangle. In fact, it can be outside the triangle, as in the case of an obtuse triangle, or it can fall at the midpoint of the hypotenuse of a right triangle. See the pictures below for examples of this.

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INCENTERThe incenter is the point at which the angle bisectors of a triangle intersect. It is the center of the circle that can be inscribed in the triangle.

The incenter is the last triangle center we will be investigating. It is the point forming the origin of a circle inscribed inside the triangle. Like the centroid, the incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle. The radius of the circle is obtained by dropping a perpendicular from the incenter to any of the triangle sides. It is pictured above as the red dashed line.

To see that the incenter is in fact always inside the triangle, let's take a look at an obtuse triangle and a right triangle.

EXAMPLE:Kurt is trying to label an altitude, median, and angle bisector for this triangle.

How can Kurt determine which lines are the altitudes, medians, or angle bisectors in a triangle?

Kurt’s Solution

I can determine which line is an altitude because an altitude is the perpendicular segment from a vertex to its opposite side. The altitude is the same as the height of a triangle; therefore, the

altitude of ABC is AD.

I can determine which line is a median by locating the segment that has been drawn from a vertex of the triangle to the midpoint of its opposite side. Since point F is a midpoint, I can determine

that the median of ABC is CF.

An angle bisector is a segment drawn from a vertex, which cuts the vertex angle in half. I can determine from ABC that the angle bisector is

BE.

REFLECTING:1. What is another name for the height of a triangle?

Altitude

2. How many medians are found in every triangle?

All triangles have three sides and three vertices, therefore a triangle will have three medians.

3. Are the altitudes, medians and angle bisector of a triangle ever the same line?

In general, altitudes, medians, and angle bisectors are different segments. In certain triangles, though, they can be the same segments. In Figure, the altitude drawn from the vertex angle of an isosceles triangle can be proven to be a median as well as an angle bisector.

PRACTISINGFill in the blank with a word that will make each statement true.

a) A ____________ is a line segment drawn from a vertex to the midpoint of its opposite side.

b) An angle _____________ cuts in half the vertex angle.

c) ______________ are sometimes the same as the side of the triangle or can sometimes meet an extended base outside the triangle.