Exercises: Similarity of triangles

Two triangles are similar if their angles are equal. Check if 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 $180^\circ$.

Calculate the scale factor $k$ that converts the small triangle into the large triangle, given that they are similar. Once $k$ is found, indicate how much the side marked with the letter $x$ must measure.

Emma has drawn two triangles in Thales's position. Given that the horizontal segments are parallel, calculate the measure of side $x$ using the proportion between their corresponding sides.

A tree casts a shadow of $12$ feet. At that same moment, a walking stick of $1,5$ feet driven vertically into the ground casts a shadow of $2$ feet. What is the height of the tree? Explain why the triangles formed by the tree and the stick with their shadows are similar.

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

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

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

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

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

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

Two 7th grade students, Ashley and Marcus, are checking if two triangular sails from their dinghies are similar. Ashley's sail has two angles measuring $65^\circ$ and $45^\circ$. Marcus's sail has one angle of $45^\circ$ and another of $70^\circ$.

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

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

To measure the height of a basketball hoop, a group of students uses a pole of $1,5$ feet placed vertically. At the same instant that the shadow of the pole measures $1,2$ feet, the shadow of the hoop projected on the ground measures $2,4$ feet. Calculate the height of the hoop taking advantage of the similarity of the triangles formed by the sun.

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

(Note: At the base, the small segment measures $4\text{ m}$ and the total $10\text{ m}$. The parallel reinforcement measures $5\text{ m}$. Find $x$, the remaining base, and $y$, the right side of the reinforcement).

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

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

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

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

Calculate the distance between points $B$ and $E$ in the following figure, given that segments $BC$ and $DE$ are parallel. The known data are: $AB = 6\text{ cm}$, $BC = 5\text{ cm}$ and $DE = 12,5\text{ cm}$.

The town council wants to install a wheelchair access ramp. The original design is a right triangle with a base of $4$ feet and a height of $0,8$ feet. By regulation, they must extend the base to $6$ feet while maintaining the same inclination (angle).

a) What height will the new ramp have?

b) If $4,08$ feet of rolling surface (hypotenuse) were needed for the original ramp, how many feet will be needed for the extended ramp?

Daniel wants to measure the depth of a circular well of $1,6$ feet in diameter. To do this, he moves $0,8$ feet away from the edge until, from a height of $1,75$ feet (his eye level), he manages to see exactly the edge of the bottom of the well. Use the similarity of triangles to find the depth of the well.

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 by citing the similarity criterion applied.

b) What is the scale factor between the second and the first?

c) If the base of the first triangle measures $6,84\text{ cm}$, what will the base of the second measure?

A group of 7th grade students wants to measure the height of the communication tower in their city. To do this, they use a vertical pole $2,5$ feet high. They place the pole in such a way that, at a given moment, the end of the tower's shadow and that of the pole coincide at the same point on the ground. If the distance between the base of the pole and the point where the shadows end is $3,2$ feet, and the distance between the base of the tower and the base of the pole is $18,4$ feet, calculate the height of the tower.

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

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

An outdoor cinema screen projects an image from a spotlight located $12$ feet away. The original screen has a surface area of $24$ $m^2$. If we move the screen closer to the projector until it is placed at exactly half the distance ($6$ feet from the spotlight), the projected image will be smaller but will keep the same shape. What will the area of the image be in this new position? Reason your answer using the relationship between the scale factor and the 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 cross at point $C$. If the measurements taken are $AB = 24$ ft, $BC = 30$ ft, and $CD = 15$ ft, determine the width of the river ($ED$) and explain which similarity criterion you have 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 feet ($m^3$)? Remember to correctly apply the relationship between the scale factor and the volume.

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

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