Exercises: Similar 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 $180^\circ$.

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 $x$.

Aoife 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.

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

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.

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?

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.

Hugo is studying for the Mathematics exam and has to determine whether two triangles are similar. The first triangle has sides measuring $4\text{ cm}$, $6\text{ cm}$, and $8\text{ cm}$, and the second has sides measuring $7\text{ cm}$, $9\text{ cm}$, and $11\text{ cm}$. Hugo writes in his notebook:\n“They are similar because if you subtract the corresponding sides, $7 - 4 = 3$, $9 - 6 = 3$, and $11 - 8 = 3$; since the difference is always the same ($3\text{ cm}$), both triangles grow at the same rate and have the same shape.”\n\na) Identify the conceptual error Hugo makes and explain in your own words why a constant difference between sides does not guarantee that two figures are similar.\nb) Check mathematically, by calculating the ratios of the corresponding sides, whether or not Hugo's triangles are similar.\nc) Write down the measurements the second triangle should have if the side corresponding to the $4\text{ cm}$ side were actually $7\text{ cm}$ so that both would be similar.

In a park in Galway, a vertical lamppost casts a $4.8\text{ m}$ shadow on the ground at the same moment that Cian, who is $1.60\text{ m}$ tall, casts a $1.20\text{ m}$ shadow.\n\na) Before doing any written calculations, observe the measurements of Cian and his shadow: is his shadow longer or shorter than his actual height? Based on this, estimate whether the height of the lamppost will be greater or less than its $4.8\text{ m}$ shadow, and write down a reasoned estimate of the approximate range of values in which its actual height should fall.\nb) Justify why the triangles formed by the lamppost with its shadow and by Cian with his are similar, citing the applicable similarity criterion.\nc) Set up the corresponding proportion, calculate the exact height of the lamppost, and check whether your estimate in part a) was accurate.

Two Year 8 students, Aoife and Seán, are checking whether two triangular sails from their dinghies are similar. Aoife's sail has two angles measuring $65^\circ$ and $45^\circ$. Seán'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).

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

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

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

b) If the perimetre of the area it covered on the ground was $120$ metres, what will the perimetre 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?

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 $DE = 12.5\text{ cm}$.

The local council wants to install a wheelchair access ramp. The original design is a right-angled triangle with a base of $4$ metres and a height of $0.8$ metres. Due to regulations, they must extend the base to $6$ metres 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 $4.08$ metres, how many metres will be required for the extended ramp?

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

The Building Regulations (Technical Guidance Document M) in Ireland regulate that access ramps for people with reduced mobility must not exceed a maximum slope nor overcome height differences greater than $1.20\text{ m}$ without including a horizontal landing. At a health centre in Galway, a straight ramp has been built. Upon inspection, the technical architect verifies that over the first $3\text{ m}$ of horizontal run, the ramp has risen $0.18\text{ m}$ in height above the ground.\n\na) Model the situation using right triangles and explain why any point along the ramp forms a triangle with the ground that is similar to the one spanning the entire ramp.\nb) If the total horizontal distance from the beginning of the ramp to the entrance door is $18\text{ m}$, calculate the total height that the ramp overcomes.\nc) With the data obtained and the indicated regulations, determine whether the ramp can be continuous or if it is mandatory to include an intermediate horizontal landing. Justify your conclusion.

A craft workshop manufactures wooden triangular frames. They have a reference pattern frame whose sides measure $5\text{ cm}$, $8\text{ cm}$, and $10\text{ cm}$. The workshop receives an order to design two new triangular frames, frame $A$ and frame $B$, which must meet the condition of having two sides measuring exactly $15\text{ cm}$ and $24\text{ cm}$.\n\na) Determine what the exact measurement of the third side of frame $A$ must be for it to be similar to the pattern frame. State which similarity criterion you applied and what the scale factor $k$ relating them is.\nb) Propose a possible integer measurement for the third side of frame $B$ so that it forms a geometrically valid triangle (remember that the sum of any two sides must always be greater than the third), but is NOT similar to the pattern frame.\nc) Prove in detail why the frame $B$ you proposed is not similar to the pattern frame by comparing the ratios between their sides.

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 $2.5$ metres 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 $3.2$ metres, and the distance from the base of the tower to the base of the pole is $18.4$ metres, calculate the height of the tower.

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.

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.

An outdoor cinema screen projects an image from a light source located $12$ metres 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 ($6$ metres 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 centre 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 metres ($m^3$)? 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.

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).

On a skateboard track, a jump ramp is built whose side section is a right triangle. The altitude drawn to the hypotenuse (the ground) divides it into two projected segments measuring $m = 1.8\text{ m}$ and $n = 3.2\text{ m}$. We wish to calculate the length of the uphill ramp, which corresponds to the shorter leg $c$ adjacent to projection $m$.\n\nTo solve it, two designers propose different methods:\n- Method 1 (Aoife): First calculate the altitude $h$ using the geometric mean theorem ($h^2 = m \times n$) and then apply the Pythagorean theorem to the smaller right triangle to find $c = \sqrt{h^2 + m^2}$.\n- Method 2 (Seán): Directly apply the right triangle leg theorem derived from triangle similarity, according to which $c^2 = m \times (m + n)$, solving for $c$ in a single step.\n\na) Apply both methods step by step with the given data and check that exactly the same numerical value is obtained for length $c$.\nb) Prove algebraically that Aoife's formula ($c^2 = h^2 + m^2$) is exactly equivalent to Seán's ($c^2 = m \times (m + n)$) by substituting the equality $h^2 = m \times n$ into it and factoring out the common term.\nc) If the starting measurements resulted in non-exact square roots and you could not use a calculator, justify which of the two methods is more advisable to avoid carrying over rounding errors.

A pinhole camera (camera obscura) relies on a tiny hole through which light rays enter in straight lines: by crossing at the hole, they project onto the back screen an inverted image that is similar to the external object.\nA group of students builds a pinhole camera using a box with a depth of $20\text{ cm}$ (distance from the pinhole to the back screen). They want to photograph a wind turbine in a wind farm in Castilla-La Mancha whose total height is $45\text{ m}$. The camera's back screen has a maximum usable height of $15\text{ cm}$.\n\na) Model the situation by describing the two triangles formed on either side of the pinhole and prove why they are similar, indicating which angles match and which similarity criterion is satisfied.\nb) Set up the appropriate proportion and calculate the minimum distance from the wind turbine at which the camera must be placed so that its full image fits on the screen without being cropped.\nc) If for safety reasons the students are forced to move away until placing the camera at exactly three times the distance calculated in the previous section, reason without solving the entire equation again what height the projected image will now measure and how it relates to the scale factor.