Exercises


  1. A carpenter builds two triangular trusses for a construction job. The trusses must be similar to one another.

    A triangular truss is made of three pieces of wood that form a triangle.

    Which of the following tests performed on each truss would be guaranteed to yield enough information to check for similarity? 

    1. Measure any one angle.
    2. Measure any two angles.
    3. Measure all three angles.
    4. Measure any one side and any one angle.
    5. Measure any one side and any two angles.
    6. Measure any two sides and any one angle.
    7. Measure any two sides and the contained angle.
    8. Measure all three sides.
  2. Which of the following statements about isosceles triangles are true? Provide examples to support your answers.
    1. All pairs of isosceles triangles are similar.
    2. Two isosceles triangles each containing at least one side length equal to \(1\) must be similar.
    3. Two isosceles triangles each containing at least one \(20^{\circ}\) angle must be similar.
    4. Two isosceles triangles each containing at least one \(120^{\circ}\) angle must be similar.

  3. In the flag of Antigua and Barbuda (shown below), there are three similar triangles and two congruent triangles. Label points on the flag and use them to write appropriate statements of similarity and congruence using similarity rules as needed.

    The rectangular flag is split into three large triangles created by drawing lines from the top left corner and top right corner to the middle of the bottom. Inside the middle triangle are two other triangles that share a vertex with the larger middle triangle. On top of the two smaller triangles but inside the larger triangle is a sun like shape.

    Source: Antigua and Barbuda Flag - Oleg Chepurin/iStock/Getty Images

  4. Consider the following diagram:

    Right angle triangle ABC shares a vertex with right angle triangle EDC where angles B and D are 90 degrees.

    1. Prove that \(\triangle ABC \sim \triangle EDC\).
    2. If \(\dfrac{AB}{DE} = \dfrac{3}{2}\), then determine the value of the ratio \(\dfrac{CE}{AC}\).
    3. If \(CD = 16\) and \(CE = 20\), then determine the lengths of all sides of \(\triangle ABC\).
  5. On a sunny day, Clara stands beside a tall tree. Her shadow measures \(3.1\) m long on the ground. At the same time, the tree's shadow measures \(11\) m along the ground.
    1. If Clara's height is known to be \(1.7\) m, then how tall is the tree to the nearest metre?
    2. Explain why it is important that the shadows are measured at the same time.
  6. When the Sun is at a particular angle, the \(45.7\) m flagpole on Windsor, Ontario's waterfront casts a shadow \(74.1\) m long. If the Sun's rays hit the \(171\) m tall Jeddah flagpole in Saudi Arabia at the same angle, what would be the length of its shadow to the nearest metre?
  7. Mickey looks up to the top of a \(14\) ft high lamp post. Directly behind the very top of the lamp post, Mickey spots his friend Lulu standing on her balcony. If \(6.0\) ft tall Mickey is standing \(8.0\) ft from the base of the lamp post and Lulu's apartment is \(110\) ft behind the lamp post, then how high is Lulu's balcony to the nearest foot?
  8. Two university buildings are connected by a \(17\) m long above-ground walkway with a glass floor. A \(1.6\) m tall student notices that if they are at the start of the walkway and look through the glass floor at a point \(2.0\) m ahead, they are staring directly at the base of the other building. Determine how high above the ground the walkway floor is to the nearest meter.
  9. A company is planning to build a rope bridge across a river. A diagram of the river shows the start and end points of the bridge. The company has made distance measurements on one side of the river as shown in the diagram. Calculate the length of the bridge to the nearest metre.

    From the starting point, a right triangle with base 32 is drawn across the river to the end of the rope bridge. The line of the rope bridge also extends backwards from the starting point creating another right triangle with base 11 and height 17.

    Source: River - ariadnaS/iStock/Getty Images

  10. In a science class, pinhole cameras are made out of boxes that are \(34.0\) cm by \(34.0\) cm by \(21.0\) cm. In the middle of a square face, a small hole is punctured and then covered with tape. In a darkroom, the entire inside of the opposite square face is covered with photosensitive paper. Emily is using her pinhole camera to take a picture of her friend Daniel who is \(180\) cm tall. What is the closest that the front of the pinhole camera can be to Daniel so that his entire \(180\) cm frame is in the picture? Assume that the camera is not held at an angle. Give you answer to the nearest centimetre.