Similarity

A graphic designer is working on the logo for a new app. The original design is a rectangle measuring $8\text{ cm}$ wide and $12\text{ cm}$ high. For the loading screen, he needs to create a larger version that is similar to the original. If the width of the enlarged version must be $20\text{ cm}$, what will be the height of the logo on that screen?

On the floor plan of a house, the scale used is $1:50$. If the living room has a length of $12\text{ cm}$ in the drawing, calculate the actual length of this room in meters.

Two triangles are positioned in Thales's configuration. Knowing that the horizontal segments are parallel and measure $6\text{ cm}$ and $10\text{ cm}$, and that the vertical segment of the smaller triangle measures $9\text{ cm}$, calculate the value of the unknown vertical segment denoted by $x$ of the larger triangle.

In the following figure, lines $r$, $s$ and $t$ are parallel. Use Thales' theorem to calculate the unknown lengths $x$ and $y$, expressed in centimeters.

Triangle $ABC$ has sides $AB = 5\text{ cm}$, $BC = 7\text{ cm}$ and $CA = 9\text{ cm}$. Triangle $DEF$ has sides $DE = 9\text{ cm}$, $EF = 5\text{ cm}$ and $FD = 7\text{ cm}$.

1. a) Are the triangles congruent? By which criterion?
2. b) Write the congruence statement with the vertices in the correct order; that is, say which vertex of $DEF$ corresponds to $A$, to $B$ and to $C$.

Diego wants to know the distance between two trees, $A$ and $B$, on opposite shores of the lake in Chapultepec Park, without crossing it. From a point $C$ on land where he can see both trees, he does the following:

• He extends line $AC$ past $C$ to a point $D$ such that $CD = CA$.

• He extends line $BC$ past $C$ to a point $E$ such that $CE = CB$.

• He measures the distance $DE$ with a tape measure: it is $63\text{ m}$.

1. a) Why are triangles $ABC$ and $DEC$ congruent? Say which parts are equal and name the criterion.
2. b) What is the distance between trees $A$ and $B$?

Emiliano builds a kite (a papalote) $ABCD$ with two sticks: the long one, $AC$, and the short one, $BD$, which cross at point $M$. The sides measure $AB = AD = 25\text{ cm}$ and $CB = CD = 51\text{ cm}$, and the long stick is $AC = 52\text{ cm}$.

1. a) Explain why $\triangle ABC \cong \triangle ADC$ and conclude that stick $AC$ splits angle $\hat{A}$ into two equal angles.
2. b) Explain why $\triangle ABM \cong \triangle ADM$. Conclude that $BM = MD$ and that the two sticks are perpendicular.
3. c) Find the length $AM$ and the length of the short stick $BD$. (Hint: apply the Pythagorean theorem to triangles $ABM$ and $CBM$, letting $x = AM$.)
4. d) How many square centimeters of tissue paper does he need to cover the kite?

State whether the following pairs of triangles are similar and justify which criterion you used (AAA, SSS or SAS):

• Triangle 1: Angles of $40^\circ$ and $70^\circ$.

• Triangle 2: Angles of $70^\circ$ and $70^\circ$.

Observe the right-angled triangle in the figure. The altitude $h$ to the hypotenuse divides the original triangle into two smaller triangles.

1. a) Justify why the three triangles (the original and the two small ones) are similar to each other.
2. b) If the segments into which the hypotenuse is divided measure $m = 4\text{ cm}$ and $n = 9\text{ cm}$, calculate the altitude $h$ using the similarity of the two small triangles.
3. c) Calculate the area of the original triangle.

A 9th-grade student is $1.60\text{ m}$ tall. At a certain time of day, they cast a shadow on the ground of $2\text{ m}$. If right next to them, a lamppost casts a shadow of $5\text{ m}$, what is the height of the lamppost?

To calculate the height of a communication tower that casts a very long shadow, Marcos, who is $1.70\text{ m}$ tall, stands so that the tip of his shadow coincides with the tip of the tower's shadow. If Marcos is $25\text{ m}$ from the base of the tower and his shadow measures $2.5\text{ m}$, answer:

1. a) Why can we state that the triangles formed by Marcos and the tower with their respective shadows are similar? Mention the similarity criterion used.
2. b) Calculate the height of the tower.

A pizzeria offers two sizes of round pizza that are similar. The "Small" pizza has a radius of $10\text{ cm}$ and an area of approximately $314\text{ cm}^2$. If the "Family" pizza has a radius of $20\text{ cm}$, what will its area be? (Remember the relationship between the scale factor $k$ and the areas).

Two rectangles are similar. The first has a base of $12\text{ cm}$ and an area of $96\text{ cm}^2$. The second rectangle has an area of $150\text{ cm}^2$.

1. a) Calculate the scale factor between the second and first rectangle. (Hint: use the relationship between areas).
2. b) Calculate the dimensions (base and height) of the second rectangle.
3. c) What is the ratio between their perimetres?

A furniture company manufactures regular hexagonal coffee tables. They have two models: the "Mini" model has a side length of $30\text{ cm}$ and its glass top costs $\$900$. The "Gran Salón" model is similar to the previous one and has a side length of $60\text{ cm}$.

1. a) What is the scale factor between the large model and the small one?
2. b) Knowing that the price of the glass is proportional to its area, how much will the glass top of the "Gran Salón" model cost?
3. c) If $180\text{ cm}$ of metal tape is needed to border the small table, how much tape will be needed for the large table?