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 $180^\circ$.

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).

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?

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.

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 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 $DE = 12.5\text{ cm}$.

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?

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).

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.

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.

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.

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.

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.

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.