Similarity of figures
A map, the plan of a house, two containers with the same shape but different sizes: similarity is not just about triangles and rectangles. We are going to generalise it to any figure, and tie up the loose ends between length, area and volume.
Similar polygons
Look at the two similar polygons in the figure. Calculate the lengths of the three unknown sides of the second polygon, knowing that the side corresponding to…
Look at the two similar polygons in the figure. Calculate the lengths of the three unknown sides of the second polygon, knowing that the side corresponding to $3 \text{ cm}$ measures $6 \text{ cm}$.
Determine whether two rectangles are similar, knowing that the dimensions of the first are $15 \text{ cm}$ base by $10 \text{ cm}$ height, and those of the second are…
Determine whether two rectangles are similar, knowing that the dimensions of the first are $15 \text{ cm}$ base by $10 \text{ cm}$ height, and those of the second are $24 \text{ cm}$ base by $16 \text{ cm}$ height. Justify your answer.
Scales: from maps to architects' plans
A scale $1:n$ tells you how many times bigger reality is than the drawing. The drawing and reality are two similar figures with $k = n$.
On a map drawn to a scale of $1:25 000$, the straight-line distance between two villages is $8 \text{ cm}$. Calculate the actual distance between both villages in…
On a map drawn to a scale of $1:25 000$, the straight-line distance between two villages is $8 \text{ cm}$. Calculate the actual distance between both villages in kilometres.
The actual straight-line distance between two cities is $120 \text{ km}$. If we want to represent them on a map to a scale of $1:500 000$, how far apart will they be on…
The actual straight-line distance between two cities is $120 \text{ km}$. If we want to represent them on a map to a scale of $1:500 000$, how far apart will they be on the map, in centimetres?
In a pizzeria, a small pizza with a diameter of $20 \text{ cm}$ costs $6 \text{ \euro}$. If the price is strictly proportional to the area of the pizza, how much should…
In a pizzeria, a small pizza with a diameter of $20 \text{ cm}$ costs $6 \text{ \euro}$. If the price is strictly proportional to the area of the pizza, how much should a family pizza with a diameter of $30 \text{ cm}$ cost?
Perimeter, area and volume according to $k$
Two triangles are similar with a scale factor of $k = 3$. If the perimeter of the smaller triangle is $14 \text{ cm}$, what is the perimeter of the larger triangle?
Two triangles are similar with a scale factor of $k = 3$. If the perimeter of the smaller triangle is $14 \text{ cm}$, what is the perimeter of the larger triangle?
The area of a square is $16 \text{ cm}^2$. If we construct a similar square using a scale factor of $k = 2,5$, what will the area of the new square be?
The area of a square is $16 \text{ cm}^2$. If we construct a similar square using a scale factor of $k = 2,5$, what will the area of the new square be?
The areas of two regular pentagons are $25 \text{ cm}^2$ and $100 \text{ cm}^2$. What is the scale factor of the sides of the larger pentagon relative to the sides of…
The areas of two regular pentagons are $25 \text{ cm}^2$ and $100 \text{ cm}^2$. What is the scale factor of the sides of the larger pentagon relative to the sides of the smaller pentagon?
From faces to similar solids
The same rule holds for whole geometric solids: two boxes, two tins or two pieces with the same shape but different sizes are similar solids, and their volume is multiplied by $k^3$.
A packing box in the shape of a cuboid has a volume of $50 \text{ dm}^3$. If we make a similar box whose dimensions are exactly double those of the original, what will…
A packing box in the shape of a cuboid has a volume of $50 \text{ dm}^3$. If we make a similar box whose dimensions are exactly double those of the original, what will its volume be?
When enlarging a rectangular photograph in an editing program, its area increased by $69\%$. Determine the scale factor $k$ applied in the enlargement and calculate the…
When enlarging a rectangular photograph in an editing program, its area increased by $69\%$. Determine the scale factor $k$ applied in the enlargement and calculate the percentage by which the perimeter of the photograph has increased.
Comparing the ratios of area, perimeter and volume with each other
Sometimes the information you are given is not $k$ directly, but the ratio between two areas, two perimeters or two volumes. You have to work backwards: from area to $k$ using the square root, from volume to $k$ using the cube root, and from there to whatever you are asked for.
Two spheres are similar. The surface area of the larger sphere is $9$ times the surface area of the smaller one. If the volume of the smaller sphere is $4 \text{ cm}^3$,…
Two spheres are similar. The surface area of the larger sphere is $9$ times the surface area of the smaller one. If the volume of the smaller sphere is $4 \text{ cm}^3$, what is the volume of the larger sphere?
A titanium engineering component weighs $4,5 \text{ kg}$. A geometrically similar component of reduced size is to be manufactured, which must weigh exactly…
A titanium engineering component weighs $4,5 \text{ kg}$. A geometrically similar component of reduced size is to be manufactured, which must weigh exactly $500 \text{ grams}$. If the original component has a maximum length of $30 \text{ cm}$, what will the maximum length of the new reduced component be?
A company manufactures similar cylindrical tins. Tin A requires $150 \text{ cm}^2$ of tinplate for its manufacture and contains $300 \text{ ml}$ of juice. If tin B…
A company manufactures similar cylindrical tins. Tin A requires $150 \text{ cm}^2$ of tinplate for its manufacture and contains $300 \text{ ml}$ of juice. If tin B requires $540 \text{ cm}^2$ of tinplate to be manufactured, how many millilitres of juice does it contain?
Dynamic geometry lab: transform and check
A tile design must repeat a motif without distorting it. Using GeoGebra or another dynamic geometry application, construct the triangle $A(1,1)$, $B(4,1)$, $C(1,3)$ and measure its sides, perimeter and area. Its base is $3$, its height is $2$ and its area is $3$ square units.
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Define the vector $(3,1)$ and use the translation tool on the whole triangle. Check that $A'(4,2)$, $B'(7,2)$, $C'(4,4)$. Predict which measurements will be preserved before checking in the application.
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On the original triangle, apply a 90° anticlockwise rotation about the centre $O(0,0)$. You should get $(-1,1)$, $(-1,4)$, $(-3,1)$. Check the distances to the centre and the lengths.
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Reflect the original in the X axis using the axial symmetry tool (reflection): $(1,-1)$, $(4,-1)$, $(1,-3)$. Lengths, angles and area are preserved, but the orientation is reversed.
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Apply to the original an enlargement with scale factor 2 and centre O: $(2,2)$, $(8,2)$, $(2,6)$. The lengths and the perimeter are doubled; the area goes from $3$ to $12$. This is similarity, not an isometry.
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Drag C to $(1,4)$, keeping A and B fixed. The images should update automatically. The original area is now $4{,}5$ and the enlarged one is $18$. Check that the three isometries preserve the new area. If the images do not move, you had drawn independent copies: rebuild the transformation linked to the original.
The diagram shows a translation: all the vertices are shifted by the same vector.
Conjecture and justification: isometries preserve distances. Checking with several drawings provides evidence, but it does not prove every case. In a translation, the differences between coordinates do not change: $(x_2+u)-(x_1+u)=x_2-x_1$ and the same for Y; that is why the distance formula keeps its value. In an enlargement with positive scale factor k, those differences are multiplied by k and so are the distances.
Create and debug instructions: write a procedure to transform any point $P(x,y)$. Translate: $(x+3,y+1)$; rotate 90° anticlockwise about O: $(-y,x)$; reflect in X: $(x,-y)$. A classmate suggests $(y,x)$ for the rotation: test it with $(1,0)$ and $(0,1)$ and explain why the second case reveals the error. Modify the algorithm for a clockwise rotation, $(y,-x)$, and check it with both points.
Cooperation: one person constructs, another measures and another looks for counterexamples; take turns. If a conjecture fails, keep the case and explain which condition was missing. Listen to all the proposals and share out the use of the tool without attributing skills to gender. To settle a disagreement, agree on a test case. If you get stuck, identify the transformation and try it out on a single point before going back to the triangle.
Mathematics, contributions and recognition
Look in a scientific or educational source for a contribution made by a woman mathematician to geometry, symmetry or their applications. Note down her name, the problem she studied and the source; explain the idea in your own words and relate it to the lab. Compare which names you knew beforehand and which you did not: historical visibility does not prove differences in ability by gender. Include the contributions of everyone on the team in the presentation.