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Lesson · Grade 9 · Mathematics

Similar triangles: same angles, sides in proportion

  • Grade 9
  • Mathematics
  • 2 min 4 s
  • British Columbia
  • BC9-MA-24

A 3, 4, 5 triangle is compared with a 6, 8, 10 triangle: their corresponding angles match, a copy of the small one slides over and doubles in size to fit exactly, and every pair of corresponding sides divides to 2, so they are similar with scale factor 2. A 6, 8, 12 triangle then fails the same test (12 ÷ 5 = 2.4, and it has an angle of about 117° where the small one has its right angle), and a turned copy of the big triangle shows the common mistake of pairing sides by position (8 ÷ 3 ≈ 2.67), fixed by pairing the sides opposite equal angles.

Concept: Similar triangles: matching sides are in the same ratio

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In one sentence

The question this lesson answers

When are two triangles similar: the same shape, but maybe a different size?

This triangle has sides 3, 4 and 5 centimetres, and this one has 6, 8 and 10.

Transcript

What the lesson says

When are two triangles similar: the same shape, but maybe a different size?

This triangle has sides 3, 4 and 5 centimetres, and this one has 6, 8 and 10.

Corresponding angles, the ones in matching positions, are equal here: a right angle, about 37 degrees and about 53 degrees.

Slide the small one over and enlarge it to twice the size: it fits exactly.

Now divide corresponding sides to find their ratios: 6 divided by 3 is 2, 8 divided by 4 is 2, and 10 divided by 5 is 2.

All three ratios are equal, so the sides are in proportion. The scale factor is 2: every length is multiplied by 2.

Same angles and sides in proportion, so these triangles are similar.

Now test the small triangle against one with sides 6, 8 and 12.

Pair the sides from shortest to longest: 6 divided by 3 is 2, 8 divided by 4 is 2, but 12 divided by 5 is 2.4.

The ratios are not all equal, so the triangles are not similar. The angles differ too: where the small one has a right angle, this one has about 117 degrees.

A common mistake is pairing sides by position. With the big triangle turned, the upright sides give 8 divided by 3, about 2.67, not 2.

Instead, pair the sides opposite, or across from, equal angles: 3 with 6, 4 with 8, and 5 with 10. Every ratio is 2, so they are similar.

For triangles and other polygons, similar means equal corresponding angles and corresponding sides in proportion.

Curriculum

Where this fits

British Columbia, Grade 9

Mathematics, Geometry and measurement: spatial proportional reasoningBC9-MA-24

Previous: Reading a Map Scale: From Centimetres to KilometresNext: How Tall Is the Tree? Similar Triangles and Shadows

British Columbia, Grade 12

Apprenticeship Mathematics 12, Triangles: similar triangles: including right-angle trigonometryBC12-MA-005

This lesson is also listed in Grade 12.

The outcomes and strands follow the British Columbia curriculum.

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