Shape and Space 11. Similarity and Congruency Mr F’s Maths Notes Shape and Space 11. Similarity and Congruency

11. Similarity and Congruency 1. If two shapes are Congruent, what does that mean? When mathematicians say that two shapes are congruent, it is just a posh, complicated way of saying that those shapes are IDENTICAL They may have been flipped upside down and rotated around, but they are still exactly the same shape and the same size

2. Congruent Triangles Because triangles only have three sides, and we know that all their interior angles must add up to 1800, we don't actually need to know every single piece of information about two triangles to be able to say that they are congruent (identical). There are 4 sets of criteria, and if a pair of triangles match any of these, then we can say for definite that they are the exact same triangle, and so they are congruent! 1. Three Sides equal (SSS) The lengths of all three sides are given in the question, and they are the same for both triangles

2. Two Sides and the included Angle equal (SAS) Two sides are the same length, and the angle in between those two sides is the same size! 3. Two Angles and a corresponding Side equal (AAS) Two angles are equal, and so too is a side in the same position relative to those two angles! 4. Right angle, Hypotenuse and Side (RHS) The triangle has a right angle, and you know the length of the hypotenuse and another side!

4 cm 120o 35o 10 cm 8 cm 4 cm 120o 8 cm 13 cm 13 cm 5 cm 5 cm 20o 3. Examples When answering questions on congruent triangles, you must quote one of the above four conditions if you believe a pair of triangles to be congruent: 4 cm 120o 35o 10 cm 8 cm 4 cm 120o These two triangles are congruent because of AAS 8 cm 13 cm 13 cm 5 cm 5 cm 20o 12 cm These two triangles are congruent because of RHS

4. If two shapes are Similar, what does that mean? Unfortunately, when mathematicians says that two objects are similar, they do not mean that they look a bit a like They mean that one object is an enlargement of the other Technically, to get from one object to the other you must multiply (or divide) every single length by the same number Just like when we dealt with Enlargement, this number is called the Scale Factor!

C B A 5. Using Length Scale Factors If we are told that two object are similar, and we can work out the scale factor, then it is possible to work out a lot of unknown information about both objects Example - These three shapes are similar. Find the missing values To Find p: 18 cm Okay, so we know the shapes are similar, so let’s work out the scale factor between rectangles A and B: p cm 4 cm q cm C So, we must enlarge every length on Rectangle A by a scale factor of 3 to get the lengths of Rectangle B. So, our missing length must be: B 48 cm A 16 cm To Find q: So now we have our scale factor, it’s dead easy to work out our missing length: Okay, so now let’s work out how to get from Rectangle A to Rectangle C

6. Similar Triangles For any other shape to be similar, all angles must be the same and all matching sides must be in proportion But… because triangles are funny, all you need for similarity between two triangles is for all three angles to be the same. Then you can be sure one triangle is an enlargement of the other Example (a) How do you know these two triangles are similar? (b) Find the unknown lengths Part (a) Two triangles are similar if all their angles are the same… Well… if you work out the missing angle in the yellow triangle it is 250, and the missing angle in the green triangle is… 350 So… all the angles are the same, so the triangles are similar! And because they are similar, we can work out the scale factor, using our matching sides between the 1200 and the 350… 3.4cm 350 X 2.5cm 250 1200 7.5cm Y 1200 6.3cm To Find X To Find Y So, to get from one triangle to the other, we either multiply or divide by 3!

60 cm ? 40 cm 20.25 litres 7. Area and Volume Factors It is also possible for 3D shapes to be similar. If we can work out the scale factor between their lengths of sides, we can also say that: Area Factor = Scale Factor2 Volume Factor = Scale Factor3 Okay, before we can do anything we need to work out the length scale factor in exactly the same way as we always do: Example - These two containers are similar. Work out the volume of water the smaller one can hold So, if our length scale factor = 1.5 Volume Scale Factor = 1.53 = 3.375 ? 40 cm 20.25 litres 60 cm So now we know how to get from the big container to the small container, so we can work out its volume:

Good luck with your revision!