Exercises: 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.

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?

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 $70^\circ$.

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).

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 $C$.

We know that $\triangle PQR \cong \triangle XYZ$, with the correspondence $P \leftrightarrow X$, $Q \leftrightarrow Y$, $R \leftrightarrow Z$. In triangle $PQR$ we measured $PQ = 8\text{ cm}$, $QR = 11\text{ cm}$, $\hat{P} = 48^\circ$ and $\hat{Q} = 67^\circ$.

Without measuring triangle $XYZ$, state the values of $XY$, $YZ$, $\hat{X}$, $\hat{Y}$ and $\hat{Z}$.

In each case you are given the parts that are equal in two triangles. Decide whether those parts guarantee that the triangles are congruent and, if so, by which criterion (SSS, SAS or ASA):

1. a) Both have a side of $6\text{ cm}$, another side of $10\text{ cm}$, and the angle between those two sides measures $35^\circ$.
2. b) Both have a side of $12\text{ cm}$, and the angles at the ends of that side measure $40^\circ$ and $75^\circ$.
3. c) Both have angles of $30^\circ$, $60^\circ$ and $90^\circ$.

A graphic designer is working on the logo for a new clothing brand. The original design is a quadrilateral with side lengths $6\text{ cm}$, $8\text{ cm}$, $9\text{ cm}$ and $12\text{ cm}$. The client asks for a larger version for a billboard, where the side that previously measured $9\text{ cm}$ must now measure $45\text{ cm}$.

1. a) Calculate the scale factor $k$ between the billboard logo and the original.
2. b) Determine the lengths of the other three sides in the billboard logo.
3. c) If the perimetre of the original logo is $35\text{ cm}$, what will be the perimetre of the billboard logo without adding its sides one by one?

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.

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.

A movie theater projector is located $6\text{ meters}$ from a screen. The beam of light projects an image that has a height of $2.4\text{ meters}$ on the screen.

1. a) If we place an obstacle of height $40\text{ cm}$ parallel to the screen to create a shadow play, at what distance from the projector should we place it so that its shadow covers the entire height of the screen?
2. b) If we move the projector further back to $9\text{ meters}$ from the screen, what will be the new height of the projected image?

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?

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 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?

In the steel truss of a pedestrian bridge in Villahermosa, the vertical bar $CD$ is perpendicular to the beam $AB$ and meets it at its midpoint $D$. We know that $AD = DB = 4\text{ m}$ and $CD = 3\text{ m}$.

1. a) Explain why $\triangle ADC \cong \triangle BDC$ and name the criterion.
2. b) Find the length of bars $AC$ and $BC$.
3. c) If angle $\widehat{CAD}$ measures about $37^\circ$, what are the measures of $\widehat{CBD}$ and $\widehat{ACB}$?

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 $B$?

Using a straightedge and compass, Ximena is going to construct a triangle with sides of $6\text{ cm}$, $8\text{ cm}$ and $10\text{ cm}$.

1. a) Describe the construction step by step, starting with the $10\text{ cm}$ side.
2. b) Use the Pythagorean theorem to check what kind of triangle she gets, classified by its angles.
3. c) Her classmate Mateo constructs a triangle with the same measurements, but he starts with the $6\text{ cm}$ side. Will his triangle be congruent to Ximena's? By which criterion?

An architect is designing the facade of a modern auditorium. He has drawn a main right-angled triangle and, to improve acoustics, needs to draw the altitude to the hypotenuse. This altitude divides the original triangle into two smaller triangles. Knowing that the legs of the original triangle measure $18\text{ m}$ and $24\text{ m}$, use triangle similarity to calculate:

1. a) The length of the altitude drawn to the hypotenuse.
2. b) The lengths of the two segments into which the hypotenuse is divided.

A 3D printing company has created a scale model of a circular square. In the model, the area of the square is $450\text{ cm}^2$. The client requests a second model where the perimetre is $20\%$ larger than that of the first model.

1. a) What will be the area of the square in this second model?
2. b) If the actual square has a radius of $30\text{ m}$, what scale was used in the first model? (Express the scale in the format $1:n$).

To measure the height of a cooling tower at a power station, a team of engineers uses a mirror placed on the ground. The engineer stands such that, looking into the mirror, he sees the top of the tower reflected. The mirror is $45\text{ m}$ from the base of the tower. The engineer, whose eyes are at a height of $1.75\text{ m}$, is $1.5\text{ m}$ from the mirror. Between the mirror and the tower there is a safety fence of height $5\text{ m}$.

What is the minimum distance from the mirror that the fence must be positioned so that it does not block the reflection of the top of the tower from the engineer's eyes?

Two cities, A and B, are separated by a river with parallel banks. To calculate the distance between a point $P$ on the bank of city A and a point $Q$ on the bank of city B (located directly opposite), a surveyor travels $60\text{ m}$ along the bank of city A to a point $R$. From $R$, he walks another $20\text{ m}$ in the same direction to a point $S$. At $S$, he turns $90^\circ$ and moves away from the river until points $Q$ and $R$ are aligned with his current position ($T$). If the distance $ST$ is $15\text{ m}$:

1. a) Calculate the width of the river ($PQ$).
2. b) Calculate the straight-line distance between the surveyor at point $T$ and point $Q$ on the other side of the river.

A drone projects a conical beam of light onto level ground. At a height of $10\text{ m}$, the beam illuminates a circle of radius $4\text{ m}$. The drone descends to a height where the illuminated area is reduced to exactly one quarter of the original area.

Draw the diagram in your notebook.

1. a) What height is the drone at now?
2. b) If at the center of the illuminated area there is an object $1.5\text{ m}$ high, what will be the radius of the shadow cast by this object when the drone was at a height of $10\text{ m}$?

On a map of a forested area, a fire has affected a trapezium-shaped surface. The bases of the trapezium on the map measure $4\text{ cm}$ and $7\text{ cm}$, and its height is $3\text{ cm}$. It is known that the actual area affected is $16.5\text{ hectares}$ ($1\text{ ha} = 10\,000\text{ m}^2$).

1. a) Calculate the scale of the map.
2. b) If it is decided to reforest an adjacent area which on the map is a triangle similar to the one formed by extending the non-parallel sides of the trapezium, and whose base is the smaller base of the trapezium, what will be the actual area of this reforestation zone?

A slide is projected onto a screen located at a distance of $6\text{ m}$. The size of the projected image is $1.8\text{ m}$ high by $2.4\text{ m}$ wide.

1. a) If the original slide has dimensions of $24\text{ mm}$ by $32\text{ mm}$, at what distance from the slide is the projector's light source located?
2. b) If we move the screen so that it is $9\text{ m}$ from the projector, by how many times will the area of the image increase compared to the previous projection?

Consider an isosceles triangle $ABC$ where the unequal side $BC$ measures $12\text{ cm}$ and the equal sides measure $10\text{ cm}$ each. Inscribed in it is a square whose base lies on side $BC$ of the triangle and its other two vertices touch sides $AB$ and $AC$.

Use the similarity between the original triangle and the small triangle that lies above the square to calculate the side length of the square.

To find how far away a buoy $P$ is in the Bacalar lagoon, two people stand at points $A$ and $B$ on the shore, $120\text{ m}$ apart. They measure the angles $\widehat{PAB} = 50^\circ$ and $\widehat{PBA} = 70^\circ$. Then, on dry land on the other side of line $AB$, they mark a point $Q$ such that $\widehat{QAB} = 50^\circ$ and $\widehat{QBA} = 70^\circ$, and they measure $AQ = 130\text{ m}$ with a tape measure.

1. a) Explain why $\triangle ABP \cong \triangle ABQ$ and name the criterion.
2. b) How far is the buoy from point $A$?
3. c) Using triangle $ABQ$ and trigonometry, find the distance from the buoy to the shore (the perpendicular distance from $P$ to line $AB$). Round to the nearest tenth.
4. d) Why wouldn't it have been enough to copy on land a triangle with the same three angles as $ABP$?

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?