Exercises


  1. In each part, determine if it is possible for the two triangles described to be similar.
    1. Two of the angles in \(\triangle ABC\) have measures of \(60^{\circ}\) and \(70^{\circ}\). Two angles in \(\triangle EFG\) have measures of \(50^{\circ}\) and \(80^{\circ}\).
    2. Two of the angles in \(\triangle XYZ\) have measures of \(45^{\circ}\) and \(75^{\circ}\). Two angles in \(\triangle PQR\) have measures of \(60^{\circ}\) and \(45^{\circ}\).
  2. The side lengths of \(\triangle ABC\) are \(3\) cm, \(5\) cm, and \(6\) cm. The side lengths of \(\triangle PQR\) are \(30\) cm, \(18\) cm, and \(36\) cm (in some order).  Is it possible that  \(\triangle ABC\) is similar to \(\triangle PQR\)? If so, determine the scale factor.
  3. Explain how you know that the two triangles in each diagram are similar.
    1. Two triangles that lie on a straight line and both have a 90 degree angle. One triangle has a 40 degree angle and the other has a 50 degree angle marked.
    2. Notice \(\overline{JP} = 3.5\) m, \(\overline{JW} = 4\) m, \(\overline{WP} = 5\) m, \(\overline{WQ} = 12.8\) m, \(\overline{BQ} = 11.2\) m, and \(\overline{BW} = 16\) m.
    3. The triangle DHE and DGF share an angle D. Angle DHE is 71 degrees and angle DFG is 71 degrees.
  4. Natalka's garden is in the shape of a right isosceles triangle as shown. The identical side lengths are \(3.4\) m long.

    Garden - Good_Stock/iStock/Getty Images

    If the garden is enlarged to a similar triangle with a scale factor of \(2\), then what is the area of the enlarged garden?
  5. The following triangle is divided into nine equilateral triangles with side length \(1\) as shown in the diagram.
    1. You can find many triangles of different sizes in the diagram above. Determine the total number of similar triangles found in the diagram.
    2. Show that all triangles found in this diagram are similar.
  6. In each part, locate a pair of similar triangles in the diagram and use this fact to find the value of \(x\).
    1. We are given \(\overline{MC} = 3\) cm, \(\overline{CE} = 6\) cm, \(\overline{DE} = 15\) cm, and \(\overline{BC} = x\) cm.
      Triangle MBC and triangle MED share angle M. Lengths MC equals MB and CE equals BD.
    2. We are given  \(\overline{ZY} = 5\) cm, \(\overline{WY} = 3\) cm, \(\overline{VY} = 3\) cm, and \(\overline{UV} = x\) cm.
      Triangle YZW and VZU share angle Z. Both triangles have right angles at angle Y and V.
    3. We are given \(\overline{PQ} = 4\) m, \(\overline{QS} = 3\) m, \(\overline{QR} = 2\) m, and \(\overline{ST} = x\) m.
      Triangle PQS and triangle PRT share angle P. There are right angles as angle Q and angle R.
  7. When a \(1.2\) m tall person stands \(20\) m from the base of a lamppost, the shadow of the person is \(2.5\) m long. 

    Sources: Street Post - S-S-S/iStock/Getty Images; Person - leremy/iStock/Getty Images

    Use similar triangles to determine the height of the lamppost.
  8. A laser is located at point \(L\) in a rectangular room \(ABCD\) as shown.
    Rectangle ABCD where point P is on the line segment AB. Angle APL and angle BPC are equal.
    Point \(L\) is located \(2\) m below wall \(\overline{AB}\) and \(3\) m to the right of wall \(\overline{AD}\). The goal is to bounce the laser off of the mirrored wall \(\overline{AB}\), at point \(P\), so that the laser beam hits corner \(C\) after rebounding. The laser beam will bounce off of the wall so that \(\angle LPA= \angle CPB\). Describe where \(P\) must be located in order for this to occur.