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Wize High School Grade 10 Math Textbook > Congruent & Similar Polygons/Triangles

Similar Triangles

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Similar Triangles

Similar triangles are triangles that have the same shape but different sizes.

How can we tell if two triangles are similar?

  • Using angles: The corresponding angles (angles in the same relative positions) are all the same Example: ∠A=∠D\angle A=\angle D, ∠B=∠E\angle B=\angle E, ∠C=∠F\angle C=\angle F We say that △ABC ∼ △DEF\triangle ABC ~\sim~\triangle DEF.

  • Using side lengths: The corresponding sides (sides in the same relative positions) have the same proportions Example: ABRP=BCPQ=CAQR=2\dfrac{AB}{RP}=\dfrac{BC}{PQ}=\dfrac{CA}{QR}=2 We say that △ABC ∼ △RPQ\triangle ABC~\sim~\triangle RPQ.

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Example: Similar Triangles

Determine if the following triangles are similar.
Using Angles
  • Triangle 1: The 2 given angles in the first triangle are 20°20\degree and 90°90\degree. Since the sum of the interior angles in a triangle is 180°180\degree, we know that the missing angle is 70°70\degree.
  • Triangle 2: The 2 given angles in the second triangle are 20°20\degree and 90°90\degree . Similar to the first triangle, we know that the missing angle is 70°70\degree.
Matching up the same angles, we see that △ABC ∼ △QPR\triangle ABC~\sim~\triangle QPR.


Using Sides
  • Triangle 1: Using Pythagorean's theorem, we see that BC≈9.75cmBC\approx 9.75cm
  • Triangle 2: Using Pythagorean's theorem, we see that PR≈2.44PR\approx2.44
Matching up the sides:
  • longest sides: ACQR=123=4\dfrac{AC}{QR}=\dfrac{12}{3}=4
  • medium sides: BCRP≈9.752.44≈4\dfrac{BC}{RP}\approx\dfrac{9.75}{2.44}\approx4
  • shortest sides: ABQP=71.75=4\dfrac{AB}{QP}=\dfrac{7}{1.75}=4
Since the corresponding side lengths have the same proportions, we see that △ABC ∼ △QPR\triangle ABC~\sim~\triangle QPR.

Practice: Similar Triangles

If △ABC\triangle ABC is similar to △DEF\triangle DEF, solve for the missing variables w,x,y,z,p,qw, x, y, z, p, q.

Practice: Similar Triangle

Identify the similar triangles in the following diagrams, then find the missing lengths.


Practice: Similar Triangles

A flashlight that is placed on the ground is pointed at a brick wall that is 20m away. A basketball player who is 1.9m tall stands 4m in front of the flashlight. Determine the height of the shadow of the basketball player that is cast on the brickwall.