Geometry (4102).

Geometry (4102)

Isometry Iso = same Metry = measurement

Translation Rotation Reflection
3 Types of Isometries Translation Rotation Reflection

in a direction, on a plane
Moving an object, in a direction, on a plane Translations original image

Same shape, different place, same orientation
Translations t original image

Properties of translations
- Same angles - Same size - Same length of line segments - Same proportion of line lengths - Same orientation

Steps to complete translations
Step 1. Measure the arrow with a ruler in cm. Step 2. Draw 2 or 3 dotted perpendicular lines from the arrow. Step 3. From each point, draw a dotted line the same length as, and in the same direction as the arrow. Step 4. Label your new points and draw the image.

Rotating an object around the centre of rotation

Rotating an object around the
centre of rotation Rotations original image

Rotations Same shape, different place, different orientation original
image

Properties of rotations
- Same shape - Same angles - Same length of line segments - Same proportion of line lengths - Different orientation

Steps to complete a rotation
Step 1. Using a ruler, measure distance from point O to all vertices (dotted lines). Step 2. Using a compass, measure each segment and draw arcs. Step 3. Using a protractor, measure angle from point O for all vertices . Step 4. Using a ruler, connect the vertices, and draw the image.

Reading a protractor: Always read from 0° to 180°

Reading a protractor Outside numbers Inside numbers
When the point of origin is located on the right side of the baseline, read the numbers from left to right. When the point of origin is located on the left side of the baseline, read the numbers from right to left.

What is this angle?

What is this angle? Align the tip of the angle to the circle in the middle of the protractor

What is this angle? Read from 0◦ upwards

What is this angle? Read from 0◦ upwards Angle is 62.5◦

What is this angle?

What is this angle? 90◦

What is this angle? How do you know? 90◦

What is this angle?

What is this angle? How do you know? 180◦

What is this angle? 0◦ 180◦

Clockwise rotation • O

Counterclockwise rotation
• O

Example: Perform a counterclockwise rotation r of ∆ABC through 120o about point O

Step 1. Using ruler, measure distance from point O to all vertices
B C O •

Step 2. Using compass, measure each segment and draw arcs counterclockwise
B C O •

Step 3. Using protractor, measure 120o (R to L) from point O for all vertices
B A B C O •

Step 3. Using protractor, measure 120o (R to L) from point O for all vertices
B C O •

Step 4. Using ruler, connect the vertices, and label them as A', B', C'

Reflection on the line of reflection

Reflection on the line of reflection
Reflections original image

Reflections Same shape, different place, different orientation
original image

Properties of reflections
- Same shape - Same angles - Same length of line segments - Same proportion of line lengths - Different orientation

Steps to complete a reflection
Step 1. Draw perpendicular dotted lines from vertices to line of reflection. Step 2. Using a ruler, measure each segment. Step 3. Using the same distances, draw new points on the other side of the line. Step 4. Using a ruler, connect the vertices, and draw the image.

Example: Reflect triangle ABC in line s

Step 1. Draw dotted lines from vertices to line of reflection
Make sure they are perpendicular to the line! A B C s

Step 1. Draw dotted lines from vertices to line of reflection
B C s

Step 2. Using ruler, measure each segment
2.5 cm A B C 1.5 cm 2 cm s

Step 3. Using the same distances, draw new points on the other side of the line
2.5 cm A B C A' A 1.5 cm 2 cm s

2.5 cm A B C A' B 1.5 cm B' 2 cm s

2.5 cm A B C A' C 2 cm C' B' 1.5 cm s

Step 4. Using ruler, connect the vertices, and label them as A', B', C'

Summary of Isometry Translation Rotation Reflection t O s

Size transformations h

Proportional shape, same orientation
Size transformations h

Pg. 2.2 Enlarge quadrilateral ABCD under size transformation h (ratio k=2)

Enlarge quadrilateral ABCD under size transformation h (ratio k=2)

Enlarge quadrilateral ABCD under size transformation h (ratio k=2)
Make bigger! B D C

Step 1. Pick a point O somewhere near the shape
B D C This point can be anywhere near the shape. There are many possible places.

B D C Centre of similitude This point can be anywhere near the shape. There are many possible places.

Step 2. Draw lines from point O to each of the vertices, and beyond

Step 3. Measure the distance from point O to each of the vertices
4 cm A O B D C

4 cm A O 5 cm B D C

4 cm A O 5 cm B D C 4.5 cm

4 cm A O 5 cm B 2 cm D C 4.5 cm mOA = 4 cm mOB = 5 cm mOC = 4.5 cm mOD = 2 cm

Step 4. Multiply the measurement by the ratio k
4 cm A O B D C k = 2 4 cm x 2 = 8 cm A

Step 5. Starting from point O, measure the new distance on the line, and mark the point as A'
8 cm A A' O B D C k = 2 4 cm x 2 = 8 cm A

5 cm B D C k = 2 5 cm x 2 = 10 cm B

Step 5. Starting from point O, measure the new distance on the line, and mark the point as B'
10 cm k = 2 5 cm x 2 = 10 cm B

D C 4.5 cm B' k = 2 4.5 cm x 2 = 9 cm C

Step 5. Starting from point O, measure the new distance on the line, and mark the point as C'
B D C B' k = 2 9 cm C' 4.5 cm x 2 = 9 cm C

2 cm D C B' k = 2 C' 2 cm x 2 = 4 cm D

Step 5. Starting from point O, measure the new distance on the line, and mark the point as D'
B D C B' k = 2 D' 4 cm C' 2 cm x 2 = 4 cm D

Step 6. Connect the points to form quadrilateral A'B'C'D'
Ta-da!

So when k = 2, the object becomes twice as large...

Pg. 2.5 Reduce quadrilateral ABCD under size transformation h (ratio k=1/2)

Reduce quadrilateral ABCD under size transformation h (ratio k=1/2)

B A C D O This point can be anywhere near the shape. There are many possible places.

Step 2. Draw lines from point O to each of the vertices, and beyond

8 cm B A C D O

8 cm B A C D O 10 cm

8 cm B A C D O 10 cm 9 cm

8 cm B A C D O 10 cm 4 cm 9 cm mOA = 8 cm mOB = 10 cm mOC = 9 cm mOD = 4 cm

8 cm B A C D O k = 1/2 8 cm x 1/2 = 4 cm A

Step 5. Starting from point O, measure the new distance on the line, and mark the point as A'
4 cm B A C D A' O k = 1/2 8 cm x 1/2 = 4 cm A

C D A' O 10 cm k = 1/2 10 cm x 1/2 = 5 cm B

Step 5. Starting from point O, measure the new distance on the line, and mark the point as B'
5 cm B' k = 1/2 10 cm x 1/2 = 5 cm B

C D A' O B' 9 cm k = 1/2 9 cm x 1/2 = 4.5 cm C

Step 5. Starting from point O, measure the new distance on the line, and mark the point as C'
B A C D A' O B' 4.5 cm C' k = 1/2 9 cm x 1/2 = 4.5 cm C

C D A' O B' 4 cm C' k = 1/2 4 cm x 1/2 = 2 cm D

Step 5. Starting from point O, measure the new distance on the line, and mark the point as D'
B A C D A' O B' 2 cm D' C' k = 1/2 4 cm x 1/2 = 2 cm D

Step 6. Connect the points to form quadrilateral A'B'C'D'
Ta-da!

So when k = 1/2, the object becomes half as large...

Rules for transformations
If k < 1, the image is smaller than the original If k > 1, the image is larger than the original

What happens if k = 1?

The image is the same size as the original
What happens if k = 1? The image is the same size as the original

Steps to complete a transformation
Step 1. Pick a point O somewhere near the shape Step 2. Draw lines from point O to each of the vertices, and beyond Step 3. Measure the distance from point O to each of the vertices Step 4. Multiply the measurement by the ratio k Step 5. Starting from point O, measure the new distance on the line, and mark the point as ' Step 6. Connect the points to form the image

Properties of transformations
Same angles Same proportion of line lengths Same order of points Parallel and perpendicular lines kept

Pg Draw the image of triangle ABC under the size transformation h with centre O and ratio k = –2.

Draw the image of triangle ABC under the size transformation h with centre O and ratio k = –2.

B O

Step 2. Draw lines from each of the vertices to point O, and beyond

1.7 cm A C B O

1.7 cm A C B 3.2 cm O

1.7 cm A C B 3.2 cm O 1.2 cm mOA = 1.7 cm mOB = 3.2 cm mOC = 1.2 cm

1.7 cm A C B O k = –2 A 1.7 cm x –2 = –3.4 cm

When k is negative, the image is on the other side of point O 1.7 cm A C B O k = –2 A mOA'= 1.7 cm x –2 = –3.4 cm

Step 5. Starting from point O, measure the new distance on the other side of the line, and mark the point as A' A C B O k = –2 A' 3.4 cm A mOA'= 1.7 cm x –2 = –3.4 cm

C B 3.2 cm O k = –2 A' B mOB'= 3.2 cm x –2 = –6.4 cm

Step 5. Starting from point O, measure the new distance on the other side of the line, and mark the point as B' A C B 6.4 cm B' O k = –2 A' B mOB'= 3.2 cm x –2 = –6.4 cm

C B O 1.2 cm k = –2 A' C mOC'= 1.2 cm x –2 = –2.4 cm

Step 5. Starting from point O, measure the new distance on the other side of the line, and mark the point as C' 2.4 cm C' A C B B' O k = –2 A' C mOC'= 1.2 cm x –2 = –2.4 cm

Step 6. Connect the points to form the image
B B' O A' Yay!

Rules for transformations
If 0 < k < 1, the image is smaller than the original If k > 1, the image is larger than the original If k < 0, the image is on the other side of point O

Similar vs. Congruent triangles

proportional exactly the same Similar vs. Congruent triangles

Two triangles are congruent when there is an isometry between them

Two triangles are congruent when there is an isometry between them
Isometry = same measurement (Translation, Rotation, Reflection)

Pg Draw a triangle with one angle 35o with the two sides forming this angle 5cm and 7cm respectively

Step 1. Draw a line of 7cm (draw longer side first)

Step 2. At one end of the line, using the protractor, draw an angle of 35o
7 cm

Step 2. At one end of the line, using the protractor, draw an angle of 35o
Remember to align the end of the line with the centre of the protractor 7 cm

Step 3. Using the ruler, draw a line through the angle of 35o
7 cm

Step 4. Using the ruler, measure 5cm along this new side

Step 5. Using the ruler, connect the two lines
5 cm 35o 7 cm

Step 6. Outline the triangle
5 cm 35o 7 cm Done!

Draw a triangle with one angle 35o with the two sides forming this angle 5cm and 7cm respectively

What if you had drawn the triangle with the sides switched?
5 cm 35o 7 cm

Same triangle, but rotated That’s okay!
7 cm 35o 5 cm

Draw a triangle with one angle 35o with the two sides forming this angle 5cm and 7cm respectively
When given two side lengths and an angle, there is only one way to draw the triangle

S-A-S side side angle 5 cm 35o 7 cm When given two side lengths and an angle, there is only one way to draw the triangle

First property of congruent triangles
S-A-S 5 cm side side angle 35o 7 cm When given two side lengths and an angle, there is only one way to draw the triangle

S-A-S Property Two triangles are congruent if two sides and the contained angle of one triangle are congruent to the corresponding parts of the other

Steps to draw a S-A-S triangle
Step 1. Draw the longer side first Step 2. At one end of the line, using the protractor, draw the angle Step 3. Using the ruler, draw a line through the angle Step 4. Using the ruler, measure the second length along this new side Step 5. Using the ruler, connect the two lines Step 6. Outline the triangle

Pg. 3.1 Draw any sort of triangle with a 5cm side contained between an angle of 30o and one of 40o

Step 1. Draw a line of 5cm 5 cm

Step 2. Using the protractor, draw the angle of 30o

Step 2. Using the protractor, draw angle of 30o from one end

5 cm

Step 4. On other end of line, use the protractor to draw an angle of 40o
Remember to align the end of the line with the centre of the protractor 30o 5 cm

5 cm

Step 6. Connect the points to form the triangle
5 cm Done!

Draw any sort of triangle with a 5cm side contained between an angle of 30o and one of 40o

What if you had drawn the triangle with the angles on the other sides?
5 cm

Same triangle, but reflected That’s okay!
5 cm

Draw any sort of triangle with a 5cm side contained between an angle of 30o and one of 40o
When given a side length and two angles, there is only one way to draw the triangle

A-S-A side angle angle 30o 40o 5 cm When given a side length and two angles, there is only one way to draw the triangle

Second property of congruent triangles
A-S-A side angle angle 30o 40o 5 cm When given a side length and two angles, there is only one way to draw the triangle

A-S-A Property Two triangles are congruent if two angles and the included side of one triangle are congruent to the corresponding parts of the other

Steps to draw an A-S-A triangle
Step 1. Draw the side Step 2. Using the protractor, draw the first angle from one end Step 3. Using the ruler, draw a line through the first angle Step 4. Using the protractor, draw the second angle from the other end Step 5. Using the ruler, draw a line through the second angle Step 6. Connect the points to form the triangle

Pg. 3.6 Draw any sort of triangle with a 6cm side, a 5 cm side and a 3 cm side

Step 1. Draw a line of 6cm (draw the longest side first)

Step 2. Using ruler and compass, measure out 5 cm from one end and draw an arc

Step 3. Using ruler and compass, measure out 3 cm from the other end and draw an arc

Step 4. Make a point where the two arcs intersect
6 cm

Step 5. Draw lines connecting the three vertices
6 cm

Step 6. Outline your triangle
5 cm 3 cm 6 cm

What if you had drawn the triangle with the sides switched?
5 cm 3 cm 6 cm

Same triangle, but rotated That’s okay!
6 cm 3 cm 5 cm

Draw any sort of triangle with a 6cm side, a 5 cm side and a 3 cm side
When given three side lengths, there is only one way to draw the triangle

S-S-S side side side 5 cm 3 cm 6 cm When given three side lengths, there is only one way to draw the triangle

Third property of congruent triangles
S-S-S side side side 5 cm 3 cm 6 cm When given three side lengths, there is only one way to draw the triangle

S-S-S Property Two triangles are congruent if the three sides of one triangle are congruent to the corresponding sides of the other

Steps to draw a S-S-S triangle
Step 1. Draw a line (draw the longest side first) Step 2. Using ruler and compass, measure out the second line from one end and draw an arc Step 3. Using ruler and compass, measure out the third line from one end and draw an arc Step 4. Make a point where the two arcs intersect Step 5. Draw lines connecting the three vertices Step 6. Outline your triangle

Two triangles are similar when there is a similarity between them

Two triangles are similar when there is a similarity between them
Similarity = proportional sides and same angles

Let’s look at these triangles a bit more...
C' A C B B' O A'

Comparing these two triangles
B' C' A C B

Comparing the angles A' B' C' A C B A A' B B' C C'

Comparing the angles A A' B B' C C'
This means these triangles have the same corresponding angles

Comparing the sides mA'B' = mB'C' = mA'C' mAB mBC mAC A' mA'B' mAB A

Comparing the sides mA'B' = mB'C' = mA'C' mAB mBC mAC
This means these triangles have the same proportional sides (their ratios are constant) mA'B' = mB'C' = mA'C' mAB mBC mAC

Comparing the sides mA'B' = mB'C' = mA'C' mAB mBC mAC
Ratio of similitude mAB mBC mAC

Are these triangles similar or congruent?
B' C' A C B

Are these triangles similar or congruent?
B' C' A C B Why?

Why are these triangles similar?
B' C' A C B They have the same angles 2) The corresponding sides are proportional

First property of similar triangles
B' C' A C B If the measures of two angles of two triangles are known and the corresponding angles are congruent, the triangles are similar

A-A Property Two triangles are similar if they have
B' C' A C B Two triangles are similar if they have two congruent corresponding angles

Second property of similar triangles
B' C' 5 cm A C B 2.5 cm 2.5 cm 5 cm 3 cm 6 cm If the lengths of three sides of two triangles are known, and the lengths of the corresponding sides are proportional, the triangles are similar

S-S-S Property Two triangles are similar if the
side side side A' B' C' 5 cm A C B 2.5 cm 2.5 cm 5 cm 3 cm 6 cm Two triangles are similar if the three corresponding sides are proportional

Third property of similar triangles
B' C' 5 cm A C B 2.5 cm 3 cm 6 cm If two pairs of corresponding sides are proportional and the contained angles are congruent, the triangles are similar

S-A-S Property side side angle A' B' C' 5 cm A C B 2.5 cm 3 cm 6 cm Two triangles are similar if they have a congruent angle contained between two corresponding proportional sides

Remember: proportional exactly the same Similar vs. Congruent triangles

Pg. 4.2. Two similar triangles and the lengths of their sides
7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image

How did the original become the image?
7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image

How did the original become the image?
Transformation 7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image

Two similar triangles (transformation)
k = ? 7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image

k = 3 original image length of side (image) = 7.5 cm = 3
length of side (original) cm k = 3 7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image

k = 3 original image length of side (image) = 6 cm = 3

k = 3 original image length of side (image) = 4.5 cm = 3

Find the missing dimensions of the image
? ? 2.5 cm 1.5 cm 2 cm 4 cm original image

Step 1. Find k (ratio of similitude)
? ? 2.5 cm 1.5 cm 2 cm 4 cm original image

Step 1. Find k (ratio of similitude)
length of side (image) = 4 cm = 2 = k length of side (original) cm ? ? 2.5 cm 1.5 cm 2 cm 4 cm original image

Step 2. Multiply each original side by k to find each length in the image
length of side (image) = ? cm = 2 length of side (original) cm cross-multiply ? cm = 2 (1.5 cm) = 3 cm ? ? = 3 cm 2.5 cm 1.5 cm 2 cm 4 cm original image

Step 2. Multiply each original side by k to find each length in the image
length of side (image) = ? cm = 2 length of side (original) cm cross-multiply ? cm = 2 (2.5 cm) = 5 cm ? = 5 cm ? = 3 cm 2.5 cm 1.5 cm 2 cm 4 cm original image

Pg. 4.3 Similar triangles ABC and A'B'C'

Pg. 4.3 Similar triangles ABC and A'B'C'
Congruent corresponding angles Proportional sides A A' C' B C B'

Similar triangles ABC and A'B'C'
Since angles A and A' are congruent, sides BC and B'C' must be proportional A A' C' B C B'

Since angles B and B' are congruent, sides AC and A'C' must be proportional A A' C' B C B'

Since angles C and C' are congruent, sides AB and A'B' must be proportional A A' C' B C B'

k = mA'B' = mA'C' = mB'C' mAB mAC mBC A A' C' B C B'

Pg. 4.5 Similar triangles ADE and ABC

Similar triangles ADE and ABC

Since angle A is in both triangles, sides DE and BC are corresponding sides A D E B C

Since angle A is in both triangles, sides DE and BC are corresponding sides Since sides DE and BC are corresponding sides, they are parallel A D E B C

Since angle A is in both triangles, sides DE and BC are corresponding sides Since sides DE and BC are corresponding sides, they are parallel A Since sides DE and BC are parallel, angles D and B, and E and C are congruent D E B C

Similarity and congruency can be used on other shapes too!

Pg. 5. 3 Given the similar quadrilaterals below
Pg. 5.3 Given the similar quadrilaterals below. The ratio of similitude between quadrilaterals ABCD and A'B'C'D' is 3/2.

Same corresponding angles
Proportional corresponding sides Four-sided shapes Pg. 5.3 Given the similar quadrilaterals below. The ratio of similitude between quadrilaterals ABCD and A'B'C'D' is 3/2. = k ABCD = original A'B'C'D' = image

Given the similar quadrilaterals below
Given the similar quadrilaterals below. The ratio of similitude between quadrilaterals ABCD and A'B'C'D' is 3/2. A' A 1.9 cm 3.7 cm C' B D D' 2.3 cm 3.3 cm C k = 3/2 B'

What is the length of each of the sides of quadrilateral A'B'C'D'?
1.9 cm 3.7 cm B' B D D' 2.3 cm 3.3 cm C k = 3/2 C'

mAB = 1.9 cm = 2.85 cm = mA'B' mBC = 2.3 cm = 3.45 cm = mB'C' mCD = 3.3 cm = 4.95 cm = mC'D' mAD = 3.7 cm = 5.55 cm = mA'D' x (3/2) A' A 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.3 cm C k = 3/2 C'

mAB = 1.9 cm = 2.85 cm = mA'B' mBC = 2.3 cm = 3.45 cm = mB'C' mCD = 3.3 cm = 4.95 cm = mC'D' mAD = 3.7 cm = 5.55 cm = mA'D' x (3/2) A' A 2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.45 cm 3.3 cm C 4.95 cm k = 3/2 C'

What is the perimeter of quadrilateral A'B'C'D'?
2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.45 cm 3.3 cm C 4.95 cm k = 3/2 C'

Perimeter = Sum of all the lengths of the sides A' A 2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.45 cm 3.3 cm C 4.95 cm k = 3/2 C'

Perimeter A'B'C'D' = 2.85cm cm cm cm = 16.8 cm A' A 2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.45 cm 3.3 cm C 4.95 cm k = 3/2 C'

Perimeter A'B'C'D' = 2.85cm + 3.45cm + 5.55cm + 4.95cm
Is the ratio of the perimeters of the quadrilaterals equal to the ratio of similitude? Perimeter A'B'C'D' = 2.85cm cm cm cm = 16.8 cm A' A 2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.45 cm 3.3 cm C 4.95 cm k = 3/2 C'

Is the ratio of the perimeters of the quadrilaterals equal to the ratio of similitude?
Perimeter A'B'C'D' = 2.85cm cm cm cm = 16.8 cm Perimeter ABCD = 1.9 cm cm cm cm = 11.2 cm A' A 2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.45 cm 3.3 cm C 4.95 cm k = 3/2 C'

Is the ratio of the perimeters of the quadrilaterals equal to the ratio of similitude?
Perimeter A'B'C'D' = 16.8 cm = 3 = k Yes! Perimeter ABCD = 11.2 cm 2 A' A 2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D D' 2.3 cm 3.45 cm 3.3 cm C 4.95 cm k = 3/2 C'

Pg. 5.5 The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon?

k = 10/3 Five equal sides Pg. 5.5 The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon?

The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon?

The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon? k = 10/3 original image

The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon? Perimeter (image) = 150 cm Regular pentagon = 5 equal sides Each side of pentagon (image) = 150 cm ÷ 5 = 30 cm k = 10/3 original image

Regular pentagon = 5 equal sides Each side of pentagon (image)
The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon? Perimeter (image) = 150 cm Regular pentagon = 5 equal sides Each side of pentagon (image) = 150 cm ÷ 5 = 30 cm 30 cm k = 10/3 original image

Side length (image) = 30 cm = 10 = k Side length (original) = x cm 3
The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon? Side length (image) = 30 cm = 10 = k Side length (original) = x cm 3 30 cm k = 10/3 original image

Side length (image) = 30 cm = 10 = k Side length (original) = x cm 3
The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon? Side length (image) = 30 cm = 10 = k cross-multiply Side length (original) = x cm 3 30 cm k = 10/3 original image

The sides of two regular pentagons have a ratio of similitude equal to 10/3. The perimeter of the large pentagon is 150 cm. What is the length of a side of the small pentagon? 30 cm = 10 x cm 3 3 (30 cm) = 10 (x cm) 30 cm 90 cm = 10x 9 cm = x k = 10/3 original image

= Ratio of similitude = k
Scale diagrams = Ratio of similitude = k

Scale diagrams Size in picture Actual length

Scale diagrams

Pg. 6.3 Size of picture on paper
12 cm Bathroom Bedroom no. 3 Bedroom no. 2 Bedroom no. 1 C C Hall C C 8 cm Dining room Hall Living room Kitchen C Pg. 6.3 Size of picture on paper

Pg. 6.3 Actual plan of a house
12 m Bathroom Bedroom no. 3 Bedroom no. 2 Bedroom no. 1 C C Hall C C 8 m Dining room Hall Living room Kitchen C Pg. 6.3 Actual plan of a house

Scale 1 1,000,000 For any units (Same units for both) 1 unit
on the picture 1,000,000 units in real life Scale ,000,000 For any units (Same units for both)

House diagram Always goes from picture to actual object
Length of house in picture = 12 cm Actual length of house = 12 m Always goes from picture to actual object

House diagram Length of house in picture = 12 cm Actual length of house = 12 m Scale: 12 cm to 12 m 12 cm = 12 cm = 1 Remember: 1 m = 100 cm 12 m cm

House diagram Length of house in picture = 12 cm Actual length of house = 12 m Scale: 12 cm to 12 m 12 cm = 12 cm = 1 Remember: 1 m = 100 cm The scale of the house picture is 1 to 100 12 m cm

Living room Length of living room in picture = 7 cm Actual length of living room = ? Scale of house: 1 to = 7 cm 100 ?

Living room Length of living room in picture = 7 cm Actual length of living room = ? Scale of house: 1 to = 7 cm Picture length 100 ? Actual length

Living room Length of living room in picture = 7 cm Actual length of living room = ? Scale of house: 1 to = 7 cm cross-multiply 100 ? 100 (7 cm) = 1 (?) 700 cm = (?)

Living room Length of living room in picture = 7 cm Actual length of living room = ? Scale of house: 1 to = 7 cm cross-multiply 100 ? The living room is actually 700 cm, or 7 m, long 100 (7 cm) = 1 (?) 7000 cm = (?)

Steps to solve scale problems
To find the scale: Step 1. Measure the length(s) in the picture Step 2. Compare one picture length to its corresponding real-life length (make sure units are the same!) Step 3. Reduce the fraction to its lowest form (use calculator)

Steps to solve scale problems
To find the length in real life: Step 1. Find the scale, if it is not given Step 2. Measure the length in the picture Step 3. Compare picture to real life (make sure units are the same!) and cross-multiply

Steps to find scale and similarity
Step 1. Measure the drawing that is given Step 2. Find the corresponding height of the other picture Step 3. Using scale, find actual height of object

If Size in picture ˃ Actual length, enlargement If Size in picture = Actual length, same size If Size in picture ˂ Actual length, reduction

Geometry (4102).

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Geometry (4102).

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Presentation on theme: "Geometry (4102)."— Presentation transcript:

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About project

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