Similarity of Triangles

Two triangles are similar if their corresponding angles are equal. Check whether the following two triangles are similar by calculating the missing angle in each one:

• Triangle A has two angles measuring $45^\circ$ and $75^\circ$.

• Triangle B has two angles measuring $75^\circ$ and $60^\circ$.

Remember that the sum of the interior angles of any triangle is always $180^\circ$.

Check whether these two triangles are similar using the side-side-side criterion (SSS criterion). To do this, divide the corresponding sides (the largest by the largest, the medium by the medium, and the smallest by the smallest) and see if you get the same result.

• Triangle 1: Sides of $6$ cm, $9$ cm and $12$ cm.

• Triangle 2: Sides of $4$ cm, $6$ cm and $8$ cm.

Two right-angled triangles have an equal acute angle of $37^\circ$. Can we state that they are similar? Justify your answer using the criteria for similarity of triangles.

Two 8th grade students, Lucía and Marcos, are checking whether two triangular sails from their dinghies are similar. Lucía's sail has two angles measuring $65^\circ$ and $45^\circ$. Marcos's sail has one angle measuring $45^\circ$ and another measuring $70^\circ$.

a) Calculate the third angle of each of the sails.

b) Explain, using similarity criteria, whether the two sails have the same shape (are similar).

We have two isosceles triangles. The first has an unequal angle of $40^\circ$ and its equal sides measure $10\text{ cm}$ each. The second also has an unequal angle of $40^\circ$, but its equal sides measure $15\text{ cm}$.

a) Are these two triangles similar? Justify your answer citing the similarity test applied.

b) What is the scale factor from the second to the first?

c) If the base of the first triangle measures $6.84\text{ cm}$, how long will the base of the second one be?

Calculate the scale factor $k$ that transforms the small triangle into the large triangle, knowing that they are similar. Once $k$ is found, state the length of the side marked with the letter $x$.

Calculate the lengths of the sides of a triangle similar to another whose sides measure $5$ cm, $7$ cm and $9$ cm, if the scale factor is $k = 3$. Will the new triangle be larger or smaller than the original?

The local municipality wants to install a wheelchair access ramp. The original design is a right triangle with a base of $4$ meters and a height of $0.8$ meters. Due to regulations, they must extend the base to $6$ meters while keeping the same incline (angle).

a) What height will the new ramp have?

b) If the original ramp required a running surface (hypotenuse) of $4.08$ meters, how many meters will be required for the extended ramp?

Consider two right-angled triangles. The first has legs of $3$ cm and $4$ cm. The second triangle is similar to the first and its hypotenuse measures $15$ cm. Calculate the perimetre and the area of this second triangle. (Hint: first find the hypotenuse of the original triangle to determine the scale factor).

Marta has drawn two triangles in Thales' position. Knowing that the horizontal line segments are parallel, calculate the length of side $x$ using the proportion between their corresponding sides.

Use Thales' theorem to find the value of $x$ in the following figure, where the shelves of the triangular bookcase are parallel to the base.

In the design of a metal structure for a roof, two parallel beams are used to reinforce the main truss. Knowing that the triangles formed are in Thales' position, calculate the length of the segments marked as $x$ and $y$ in the following diagram:

(Note: On the base, the small segment measures $4\text{ m}$ and the total is $10\text{ m}$. On the inclined pitch, the small segment measures $5\text{ m}$. Find $x$, the remaining base, and $y$, the remaining segment of the inclined pitch).

Calculate the distance between points $B$ and $D$ in the following figure, knowing that segments $BC$ and $DE$ are parallel and that points $A$, $B$ and $D$ are collinear, as are $A$, $C$ and $E$. The known data are: $AB = 6\text{ cm}$, $BC = 5\text{ cm}$ and $DE = 12.5\text{ cm}$.

A drone is flying over a park at a height of $15$ meters. Its camera projects a triangular field of view on the ground. If the drone rises to a height of $45$ meters:

a) What is the scale factor between the original triangular field of view and the new one?

b) If the perimeter of the area it covered on the ground was $120$ meters, what will the perimeter of the new area be?

c) If the surface area originally covered was $800\text{ m}^2$, what will the surface area covered at the new height be?

Two transversals intersect at a point $V$. Points $A$ and $B$ are marked on one of them, and points $C$ and $D$ on the other, such that segments $AC$ and $BD$ are parallel (Thales' configuration). Given that $VA = 9$ cm, $VB = 15$ cm, and the area of the smaller triangle $VAC$ is $27$ $cm^2$, calculate the area of the quadrilateral $ABDC$, knowing that the triangles are similar.

An outdoor movie screen projects an image from a light source located $12$ meters away. The original screen has an area of $24$ $m^2$. If we move the screen closer to the projector so that it is exactly halfway ($6$ meters from the light source), the projected image will be smaller but will keep the same shape. What will the area of the image be in this new position? Explain your answer using the relationship between the scale factor and area.

To calculate the width of a river without crossing it, a surveyor places several stakes as shown in the figure. The triangles formed, $ABC$ and $CDE$, are similar because the banks are parallel and the lines of sight intersect at point $C$. If the measurements taken are $AB = 24$ m, $BC = 30$ m, and $CD = 15$ m, determine the width of the river ($ED$) and explain which similarity test you applied.

An architect presents a scale model of a cultural center in the shape of a triangular-based pyramid. The scale factor between the model and the actual building is $1:150$. If the volume of the model is $0.8$ $dm^3$, what will the volume of the actual building be, expressed in cubic meters ($m^3$)? Remember to correctly apply the relationship between the scale factor and volume.

In a triangle $ABC$, side $AB$ measures $12$ cm and side $AC$ measures $15$ cm. We draw a segment $PQ$ parallel to side $BC$, such that point $P$ lies on $AB$ and point $Q$ lies on $AC$. If segment $AP$ is one-third the length of segment $PB$, calculate the lengths of segments $AP$ and $AQ$. Justify your answer based on Thales' theorem.

A tree casts a shadow of $12$ meters. At the same moment, a stick $1.5$ meters long placed vertically in the ground casts a shadow of $2$ meters. How tall is the tree? Explain why the triangles formed by the tree and the stick with their shadows are similar.

To measure the height of a basketball hoop, a group of students uses a stick $1.5$ meters tall placed vertically. At the same moment that the shadow of the stick measures $1.2$ meters, the shadow of the basketball hoop cast on the ground measures $2.4$ meters. Calculate the height of the basketball hoop using the similarity of the triangles formed by the sun.

Daniel wants to measure the depth of a circular well with a diameter of $1.6$ meters. To do so, he steps $0.8$ meters back from the edge until, from a height of $1.75$ meters (eye level), he can see the exact edge of the bottom of the well. Use similar triangles to find the depth of the well.

A group of Year 8 students wants to measure the height of their city's communications tower. To do so, they use a vertical pole $2.5$ meters tall. They place the pole in such a way that, at a specific moment, the tip of the tower's shadow and the tip of the pole's shadow coincide at the same point on the ground. If the distance from the base of the pole to the point where the shadows end is $3.2$ meters, and the distance from the base of the tower to the base of the pole is $18.4$ meters, calculate the height of the tower.

A carpenter is designing a shelving unit reinforced with a crossbar in the shape of a right-angled triangle. In the base triangle, the altitude to the hypotenuse divides it into two segments measuring $18$ cm and $32$ cm. The carpenter needs to know the length of the two wooden slats that form the legs of the triangle in order to buy them. Calculate the length of both legs and the height of the crossbar using similarity relationships in right-angled triangles.