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5. Now the value of x can be calculated by dividing 9 by 2. x = 2 9,4 = 4,7.
6. And y = 2 × 5,5 = 11.
Exercise:
Calculate the values of sides PR and X Y in the following triangles.
Example:
Find all the missing angles in these two triangles, if possible.
1. The sides are in proportion: 42 × 1,5 = 63 , 38 × 1,5 = 57 and 34 × 1,5 = 51
2. This means that the triangles are similar: ΔEFG ΔKLM (sides in proportion)
3. So, corresponding angles are equal: L = 68° (corresponds to F)
E = 51° (corresponds to K)
G = M = 61° (sum of the angles of a triangle)
Exercise:
Find all the missing angles in these triangles:
Activity 5
To apply similarity in problems
[LO 4.4, 1.4, 3.5]
1. Are the following triangles similar?
1.1 ΔBAG with B = 90°, AG = 15cm and AB = 9cm and
ΔPOT with P = 90°, OT = 5cm and PO = 4cm.
1.2 ΔREM with R = 60° and M = 50° and
ΔSUP with U = 70° and S = 50°.
1.3 ΔLOP with P = 90°, LO = 13cm and OP = 12cm and
ΔCAT with C = 90°, AC = 16cm and CT = 12cm
2. Calculate the proportional constant in similar triangles ΔABC and ΔDEF when
AB = 36cm, EF = 12cm, C = 48° and D = 48°.
3. Two flagpoles (one longer than the other) throw shadows on the ground. The shadow of the longer pole (which is 8 m tall) is 3 m and the shorter flagpole has a 2,5 m shadow. Calculate how tall the short flagpole is.
4. Gloria is designing a logo for her school’s computer club. The design shows a computer next to a pile of books which is 50 % higher than the computer. She is photocopying the design to make it smaller. On the photocopy the computer is 18 cm high and on the original the pile of books is 54 cm high. By what factor is she making the design smaller?
| Learning outcomes(LOs) |
| LO 1 |
| Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems. |
| Assessment standards(ASs) |
| We know this when the learner : |
| 1.1 describes and illustrates the historical development of number systems in a variety of historical and cultural conte x ts (including local); |
| 1.2 recognises, uses and represent rational numbers (including very small numbers written in scientific notation), moving fle x ibly between equivalent forms in appropriate conte x ts; |
| 1.3 solves problems in conte x t, including conte x ts that may be used to build awareness of other Learning Areas, as well as human rights, social, economic and environmental issues such as: |
| 1.3.1 financial (including profit and loss, budgets, accounts, loans, simple and compound interest, hire purchase, exchange rates, commission, rentals and banking); |
| 1.3.2 measurements in Natural Sciences and Technology contexts; |
| 1.4 solves problems that involve ratio, rate and proportion (direct and indirect); |
| LO 3 |
| Space and Shape (Geometry)The learner will be able to describe and represent characteristics and relationships between two-dimensional shapes and three-dimensional objects in a variety of orientations and positions. |
| We know this when the learner : |
| 3.1 recognises, visualises and names geometric figures and solids in natural and cultural forms and geometric settings, including:3.1.1 regular and irregular polygons and polyhedra;3.1.2 spheres;3.1.3 cylinders;3.2 in contexts that include those that may be used to build awareness of social, cultural and environmental issues, describes the interrelationships of the properties of geometric figures and solids with justification, including:3.2.1 congruence and straight line geometry;3.3 uses geometry of straight lines and triangles to solve problems and to justify relationships in geometric figures;3.4 draws and/or constructs geometric figures and makes models of solids in order to investigate and compare their properties and model situations in the environment; |
| 3.5 uses transformations, congruence and similarity to investigate, describe and justify (alone and/or as a member of a group or team) properties of geometric figures and solids, including tests for similarity and congruence of triangles. |
| LO 4 |
| MeasurementThe learner will be able to use appropriate measuring units, instruments and formulae in a variety of contexts. |
| We know this when the learner : |
| 4.1 solves ratio and rate problems involving time, distance and speed; |
| 4.2 solves problems (including problems in contexts that may be used to develop awareness of human rights, social, economic, cultural and environmental issues) involving known geometric figures and solids in a range of measurement contexts by: |
| 4.2.1 measuring precisely and selecting measuring instruments appropriate to the problem; |
| 4.2.2 estimating and calculating with precision; |
| 4.2.3 selecting and using appropriate formulae and measurements; |
| 4.3 describes and illustrates the development of measuring instruments and conventions in different cultures throughout history; |
| 4.4 uses the Theorem of Pythagoras to solve problems involving missing lengths in known geometric figures and solids. |
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