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Geometry (4102) Isometry Iso = same Metry = measurement 3 Types of Isometries Translation Rotation Reflection Moving an object, in a direction, on a plane Translations original image Same shape, different place, same orientation Translations original t 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. Rotation Rotating an object around the centre of rotation Rotating an object around the centre of rotation Rotations original image Same shape, different place, different orientation Rotations 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 • When the point of origin is located on the right side of the baseline, read the numbers from left to right. Inside numbers • 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? 90◦ What is this angle? What is this angle? 180◦ 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 A O • B C Step 1. Using ruler, measure distance from point O to all vertices A O • B C Step 1. Using ruler, measure distance from point O to all vertices A O • B C Step 2. Using compass, measure each segment and draw arcs counterclockwise A O • B C Step 2. Using compass, measure each segment and draw arcs counterclockwise A O • B C Step 2. Using compass, measure each segment and draw arcs counterclockwise A O • B C Step 3. Using protractor, measure 120o (R to L) from point O for all vertices B A O • B C Step 3. Using protractor, measure 120o (R to L) from point O for all vertices C A O • B C Step 3. Using protractor, measure 120o (R to L) from point O for all vertices A A O • B C Step 4. Using ruler, connect the vertices, and label them as A', B', C' C' C A' B' O • A B C Reflection Reflection on the line of reflection Reflection on the line of reflection Reflections original image Same shape, different place, different orientation Reflections 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 A C B s Step 1. Draw dotted lines from vertices to line of reflection Make sure they are perpendicular to the line! A C B s Step 1. Draw dotted lines from vertices to line of reflection A C B s Step 1. Draw dotted lines from vertices to line of reflection A C B s Step 2. Using ruler, measure each segment A 2.5 cm 1.5 cm C B 2 cm s Step 3. Using the same distances, draw new points on the other side of the line A 2.5 cm A' 1.5 cm C B 2 cm s A Step 3. Using the same distances, draw new points on the other side of the line A 2.5 cm A' 1.5 cm C B B' 2 cm s B Step 3. Using the same distances, draw new points on the other side of the line A 2.5 cm A' 2 cm C' C B B' 1.5 cm s C Step 4. Using ruler, connect the vertices, and label them as A', B', C' A A' C' C B B' s Summary of Isometry Translation t Rotation •O Reflection 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) A B D C Enlarge quadrilateral ABCD under size transformation h (ratio k=2) A Make bigger! B D C Step 1. Pick a point O somewhere near the shape A B • D C This point can be anywhere near the shape. There are many possible places. O Step 1. Pick a point O somewhere near the shape A B • O D C This point can be anywhere near the shape. There are many possible places. Centre of similitude Step 2. Draw lines from point O to each of the vertices, and beyond A B • D C O Step 2. Draw lines from point O to each of the vertices, and beyond A B • D C O Step 2. Draw lines from point O to each of the vertices, and beyond A B • D C O Step 3. Measure the distance from point O to each of the vertices 4 cm A B • D C O Step 3. Measure the distance from point O to each of the vertices 4 cm A • 5 cm B D C O Step 3. Measure the distance from point O to each of the vertices 4 cm A • 5 cm B D C 4.5 cm O Step 3. Measure the distance from point O to each of the vertices 4 cm A • 5 cm B D C 2 cm 4.5 cm mOA = 4 cm mOB = 5 cm mOC = 4.5 cm mOD = 2 cm O Step 4. Multiply the measurement by the ratio k 4 cm A B • O D C k=2 A 4 cm x 2 = 8 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as A' 8 cm A • A' B • O D C k=2 A 4 cm x 2 = 8 cm Step 4. Multiply the measurement by the ratio k A • A' • O 5 cm B D C k=2 B 5 cm x 2 = 10 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as B' A • A' B • B' 10 cm • O D C k=2 B 5 cm x 2 = 10 cm Step 4. Multiply the measurement by the ratio k A • A' • B • B' O D C 4.5 cm k=2 C 4.5 cm x 2 = 9 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as C' A • A' B O D C • B' • C • 9 cm k=2 C' 4.5 cm x 2 = 9 cm Step 4. Multiply the measurement by the ratio k A • A' B D O 2 cm C • B' • D • k=2 C' 2 cm x 2 = 4 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as D' A • A' • B D C • B' • • D O D' 4 cm k=2 C' 2 cm x 2 = 4 cm Step 6. Connect the points to form quadrilateral A'B'C'D' A • A' B • D C • B' • • D' C' Ta-da! O 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) A B D C Step 1. Pick a point O somewhere near the shape A • O B D C 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 A • B D C O Step 2. Draw lines from point O to each of the vertices, and beyond A • B D C O Step 2. Draw lines from point O to each of the vertices, and beyond A • B D C O Step 3. Measure the distance from point O to each of the vertices 8 cm A • B D C O Step 3. Measure the distance from point O to each of the vertices 8 cm A • 10 cm B D C O Step 3. Measure the distance from point O to each of the vertices 8 cm A • 10 cm B 9 cm C D O Step 3. Measure the distance from point O to each of the vertices 8 cm A • O 10 cm B 9 cm C D 4 cm mOA = 8 cm mOB = 10 cm mOC = 9 cm mOD = 4 cm Step 4. Multiply the measurement by the ratio k 8 cm A • O B D k = 1/2 C A 8 cm x 1/2 = 4 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as A' 4 cm A • A' • O B D k = 1/2 C A 8 cm x 1/2 = 4 cm Step 4. Multiply the measurement by the ratio k A • A' • O 10 cm B D k = 1/2 C B 10 cm x 1/2 = 5 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as B' A • A' • O 5 cm • B' B D k = 1/2 C B 10 cm x 1/2 = 5 cm Step 4. Multiply the measurement by the ratio k A • A' • • O B' 9 cm B D k = 1/2 C C 9 cm x 1/2 = 4.5 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as C' A • A' • • O B' • C' 4.5 cm B D k = 1/2 C C 9 cm x 1/2 = 4.5 cm Step 4. Multiply the measurement by the ratio k A • A' • • O B' • C' 4 cm B D k = 1/2 C D 4 cm x 1/2 = 2 cm Step 5. Starting from point O, measure the new distance on the line, and mark the point as D' A • A' • B' • C' • • D' O 2 cm B D k = 1/2 C D 4 cm x 1/2 = 2 cm Step 6. Connect the points to form quadrilateral A'B'C'D' A • A' • B' • C' • • D' B D C Ta-da! O 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? 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. 2.7 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. A B C Step 1. Pick a point O somewhere near the shape A • B O C Step 2. Draw lines from each of the vertices to point O, and beyond A • B O C Step 2. Draw lines from each of the vertices to point O, and beyond A • B O C Step 2. Draw lines from each of the vertices to point O, and beyond A • B O C Step 3. Measure the distance from point O to each of the vertices 1.7 cm A • B O C Step 3. Measure the distance from point O to each of the vertices 1.7 cm A • 3.2 cm B O C Step 3. Measure the distance from point O to each of the vertices 1.7 cm A • 3.2 cm B O 1.2 cm C mOA = 1.7 cm mOB = 3.2 cm mOC = 1.2 cm Step 4. Multiply the measurement by the ratio k 1.7 cm A • B O C A k = –2 1.7 cm x –2 = –3.4 cm Step 4. Multiply the measurement by the ratio k When k is negative, the image is on the other side of point O 1.7 cm A • B O C A k = –2 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 • B O • A' A C 3.4 cm k = –2 mOA'= 1.7 cm x –2 = –3.4 cm Step 4. Multiply the measurement by the ratio k A • 3.2 cm B O • A' B C k = –2 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 6.4 cm • • B' O • A' B B C k = –2 mOB'= 3.2 cm x –2 = –6.4 cm Step 4. Multiply the measurement by the ratio k A • B O • A' C 1.2 cm C k = –2 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' • C' 2.4 cm A • • B' O • A' C B C k = –2 mOC'= 1.2 cm x –2 = –2.4 cm Step 6. Connect the points to form the image • C' A • • B' O • A' B C 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 proportional exactly the same 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. 3.4 Draw a triangle with one angle o 35 with the two sides forming this angle 5cm and 7cm respectively Step 1. Draw a line of 7cm (draw longer side first) 7 cm 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 35o 7 cm Step 4. Using the ruler, measure 5cm along this new side 5 cm 35o 7 cm 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 5 cm 35o 7 cm 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 5 cm Draw a triangle with one angle 35o with the two sides forming this angle 5cm and 7cm respectively 5 cm 35o 7 cm 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 o o an angle of 30 and one of 40 Step 1. Draw a line of 5cm 5 cm Step 2. Using the protractor, draw the angle of 30o Remember to align the end of the line with the centre of the protractor 5 cm Step 2. Using the protractor, draw angle of 30o from one end Remember to align the end of the line with the centre of the protractor 5 cm Step 3. Using the ruler, draw a line through the angle of 30o 30o 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 Step 5. Using the ruler, draw a line through the angle of 40o 30o 40o 5 cm Step 6. Connect the points to form the triangle 30o 40o 5 cm Done! Draw any sort of triangle with a 5cm side contained between an angle of 30o and one of 40o 30o 40o 5 cm What if you had drawn the triangle with the angles on the other sides? 30o 40o 5 cm Same triangle, but reflected That’s okay! 30o 40o 5 cm Draw any sort of triangle with a 5cm side contained between an angle of 30o and one of 40o 30o 40o 5 cm 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) 6 cm Step 2. Using ruler and compass, measure out 5 cm from one end and draw an arc 5 cm 6 cm Step 2. Using ruler and compass, measure out 5 cm from one end and draw an arc 5 cm 6 cm Step 3. Using ruler and compass, measure out 3 cm from the other end and draw an arc 3 cm 6 cm Step 3. Using ruler and compass, measure out 3 cm from the other end and draw an arc 3 cm 6 cm Step 4. Make a point where the two arcs intersect 6 cm Step 5. Draw lines connecting the three vertices 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 5 cm 3 cm Draw any sort of triangle with a 6cm side, a 5 cm side and a 3 cm side 5 cm 3 cm 6 cm 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 3 cm 5 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 • • B' O • A' B C Comparing these two triangles A' A B' B C C' Comparing the angles A' A B' B C C' A A' B B' C C' Comparing the angles A' A B' B C A A' B B' C C' C' This means these triangles have the same corresponding angles Comparing the sides A' mA'B' A B' mAB B C C' mA'B' = mB'C' = mA'C' mAB mBC mAC Comparing the sides A' mA'B' A B' mAB B C C' mA'B' = mB'C' = mA'C' mAB mBC mAC This means these triangles have the same proportional sides (their ratios are constant) Comparing the sides A' mA'B' A B' mAB B C C' mA'B' = mB'C' = mA'C' mAB mBC mAC Ratio of similitude Are these triangles similar or congruent? A' A B' B C C' Are these triangles similar or congruent? A' A B' B C C' Are these triangles similar or congruent? A' A B' B C C' Why? Why are these triangles similar? A' A B' B C C' 1) They have the same angles 2) The corresponding sides are proportional First property of similar triangles A' A B' B C C' If the measures of two angles of two triangles are known and the corresponding angles are congruent, the triangles are similar A-A Property angle angle A' A B' B C C' Two triangles are similar if they have two congruent corresponding angles Second property of similar triangles A' 5 cm A 2.5 B' cm 5 cm C' 6 cm 2.5 cm B C 3 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 side side side A' 5 cm A 2.5 B' cm 5 cm C' 6 cm 2.5 cm B C 3 cm Two triangles are similar if the three corresponding sides are proportional Third property of similar triangles A' 5 cm A B B' C' 6 cm 2.5 cm C 3 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' 5 cm A B B' C' 6 cm 2.5 cm C 3 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 length of side (image) = 7.5 cm = 3 length of side (original) 2.5 cm k=3 7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image length of side (image) = 6 cm = 3 length of side (original) 2 cm k=3 7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image length of side (image) = 4.5 cm = 3 length of side (original) 1.5 cm k=3 7.5 cm 4.5 cm 2.5 cm 1.5 cm 2 cm 6 cm original image Find the missing dimensions of the image ? 2.5 cm ? 1.5 cm 2 cm original 4 cm image Step 1. Find k (ratio of similitude) ? 2.5 cm ? 1.5 cm 2 cm original 4 cm image Step 1. Find k (ratio of similitude) length of side (image) = 4 cm = 2 = k length of side (original) 2 cm ? 2.5 cm ? 1.5 cm 2 cm original 4 cm 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) 1.5 cm cross-multiply ? cm = 2 (1.5 cm) = 3 cm ? 2.5 cm ? = 3 cm 1.5 cm 2 cm original 4 cm 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) 2.5 cm cross-multiply ? cm = 2 (2.5 cm) = 5 cm ? = 5 cm 2.5 cm ? = 3 cm 1.5 cm 2 cm original 4 cm image Pg. 4.3 Similar triangles ABC and A'B'C' A A' B C B' C' Pg. 4.3 Similar triangles ABC and A'B'C' Congruent corresponding angles Proportional sides A A' B C B' C' Similar triangles ABC and A'B'C' Since angles A and A' are congruent, sides BC and B'C' must be proportional A A' B C B' C' Similar triangles ABC and A'B'C' Since angles B and B' are congruent, sides AC and A'C' must be proportional A A' B C B' C' Similar triangles ABC and A'B'C' Since angles C and C' are congruent, sides AB and A'B' must be proportional A A' B C B' C' Similar triangles ABC and A'B'C' k = mA'B' = mA'C' = mB'C' mAB mAC mBC A A' B C B' C' Pg. 4.5 Similar triangles ADE and ABC A D B E C Similar triangles ADE and ABC A D B E C Similar triangles ADE and ABC A D B E C Similar triangles ADE and ABC Since angle A is in both triangles, sides DE and BC are corresponding sides A D B E C Similar triangles ADE and ABC 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 B E C Similar triangles ADE and ABC Since angle A is in both triangles, sides DE and BC are corresponding sides A D B Since sides DE and BC are corresponding sides, they are parallel Since sides DE and BC are parallel, angles D and B, and E and C are congruent E C Similarity and congruency can be used on other shapes too! Pg. 5.3 Given the similar quadrilaterals below. The ratio of similitude between quadrilaterals ABCD and A'B'C'D' is 3/2. Four-sided shapes Same corresponding angles Proportional corresponding sides 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. 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 2.3 cm C D' 3.3 cm k = 3/2 B' What is the length of each of the sides of quadrilateral A'B'C'D'? A' A 1.9 cm 3.7 cm B' B D 2.3 cm C D' 3.3 cm k = 3/2 C' What is the length of each of the sides of quadrilateral A'B'C'D'? mAB = 1.9 cm mBC = 2.3 cm mCD = 3.3 cm mAD = 3.7 cm x (3/2) = 2.85 cm = mA'B' = 3.45 cm = mB'C' = 4.95 cm = mC'D' = 5.55 cm = mA'D' A' A 1.9 cm 3.7 cm B' B D 2.3 cm C D' 3.3 cm k = 3/2 C' What is the length of each of the sides of quadrilateral A'B'C'D'? mAB = 1.9 cm mBC = 2.3 cm mCD = 3.3 cm mAD = 3.7 cm x (3/2) = 2.85 cm = mA'B' = 3.45 cm = mB'C' = 4.95 cm = mC'D' = 5.55 cm = mA'D' A' A 1.9 cm 2.85 cm 5.55 cm 3.7 cm B' B D 2.3 cm C 3.3 cm D' 3.45 cm 4.95 cm k = 3/2 C' What is the perimeter of quadrilateral A'B'C'D'? A' A 1.9 cm 2.85 cm 5.55 cm 3.7 cm B' B D 2.3 cm C 3.3 cm D' 3.45 cm 4.95 cm k = 3/2 C' What is the perimeter of quadrilateral A'B'C'D'? Perimeter = Sum of all the lengths of the sides A' A 1.9 cm 2.85 cm 5.55 cm 3.7 cm B' B D 2.3 cm C 3.3 cm D' 3.45 cm 4.95 cm k = 3/2 C' What is the perimeter of quadrilateral A'B'C'D'? Perimeter A'B'C'D' = 2.85cm + 3.45cm + 5.55cm + 4.95cm = 16.8 cm A' A 1.9 cm 2.85 cm 5.55 cm 3.7 cm B' B D 2.3 cm C 3.3 cm D' 3.45 cm 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 + 3.45cm + 5.55cm + 4.95cm = 16.8 cm A' A 1.9 cm 2.85 cm 5.55 cm 3.7 cm B' B D 2.3 cm C 3.3 cm D' 3.45 cm 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 + 3.45cm + 5.55cm + 4.95cm = 16.8 cm Perimeter ABCD = 1.9 cm + 2.3 cm + 3.3 cm + 3.7 cm = 11.2 cm A' A 2.85 cm 5.55 cm 1.9 cm 3.7 cm B' B D 2.3 cm C 3.3 cm D' 3.45 cm 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 Perimeter ABCD = 11.2 cm 2 Yes! A' A 1.9 cm 2.85 cm 5.55 cm 3.7 cm B' B D 2.3 cm C 3.3 cm D' 3.45 cm 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 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 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 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) 90 cm = 10x 10 10 30 cm 9 cm = x k = 10/3 original image Scale diagrams = Ratio of similitude =k Size in picture Actual length Scale diagrams Size in picture Actual length Size in picture : Actual length Size in picture Actual length Scale diagrams Bedroom no. 3 12 cm Bedroom Bathroom no. 2 C C Bedroom no. 1 Hall C Dining room Hall Living room Pg. 6.3 Size of picture on paper C C 8 cm Kitchen Bedroom no. 3 12 m Bedroom Bathroom no. 2 C C Bedroom no. 1 Hall C Dining room Hall Living room Pg. 6.3 Actual plan of a house C Kitchen C 8 m 1 unit on the picture 1,000,000 units in real life Scale 1 1,000,000 For any units (Same units for both) Size in picture Actual length Scale diagrams Size in picture Actual length Size in picture : Actual length Size in picture Actual length House diagram 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 Remember: 1 m = 100 cm 12 cm = 12 cm = 1 12 m 1200 cm 100 House diagram Length of house in picture = 12 cm Actual length of house = 12 m Scale: 12 cm to 12 m Remember: 1 m = 100 cm 12 cm = 12 cm = 1 The scale of the 12 m 1200 cm 100 house picture is 1 to 100 Living room Length of living room in picture = 7 cm Actual length of living room = ? Scale of house: 1 to 100 1 = 7 cm 100 ? Living room Length of living room in picture = 7 cm Actual length of living room = ? Scale of house: 1 to 1000 1 = 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 1000 1 = 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 1000 1 = 7 cm cross-multiply 100 ? The living room is 100 (7 cm) = 1 (?) actually 700 cm, or 7 m, long 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 crossmultiply 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 Size in picture Actual length Scale diagrams If Size in picture ˃ Actual length, enlargement If Size in picture = Actual length, same size If Size in picture ˂ Actual length, reduction