Criteria for Similarity, Areas and the Pythagoras Theorem: Quick Revision

A brief note on Criteria for Similarity, Areas and the Pythagoras Theorem. Read it before a test. Try each question at the end before you open its answer.

The three similarity tests

• AA: two angles of one triangle equal two angles of the other.
• SSS: the three pairs of sides are in the same ratio, shortest with shortest.
• SAS: one angle equal, and the two sides around it in the same ratio.
• Write similar triangles with their letters in matching order, then read off the side ratios.
• Shadows, a trapezium with parallel sides, and a line $DE \parallel BC$ all give similar triangles by AA.

Areas of similar triangles

• The ratio of the areas is the square of the ratio of corresponding sides.
• The same holds for corresponding altitudes.
• Sides in the ratio $2 : 3$ give areas in the ratio $4 : 9$.
• If $DE \parallel BC$, then $\dfrac{\text{ar}(ADE)}{\text{ar}(ABC)} = \left(\dfrac{AD}{AB}\right)^2$.

The Pythagoras theorem and its converse

• Right angle at B: $AC^2 = AB^2 + BC^2$. The hypotenuse is the longest side.
• Converse: if $AC^2 = AB^2 + BC^2$, the angle at B is $90^\circ$.
• If the square of the longest side is more than the other two squares added, the triangle is obtuse. If it is less, the triangle is acute.
• When BD is the perpendicular from the right angle to the hypotenuse, $BD^2 = AD \times DC$.

A worked example

Two similar triangles have matching sides of $4$ cm and $6$ cm. The smaller triangle has an area of $24$ square cm. Find the area of the larger.

1. The ratio of the sides. $\dfrac{6}{4} = \dfrac{3}{2}$.
2. Square it. The areas are in the ratio $\dfrac{9}{4}$.
3. Multiply. $24 \times \dfrac{9}{4} = 54$ square cm.

Check. The larger triangle has the larger area, as it must.

Practice with new numbers

A program to try

The program checks the three ideas on numbers: the side ratios of two similar triangles, their area ratio, and a Pythagorean triple.

Mistakes to avoid

• Using the ratio of the sides for the areas. Square it.
• Matching the sides in the wrong order. Shortest goes with shortest.
• Using the Pythagoras theorem without a right angle.
• Adding the squares when a leg is missing. Subtract instead.

Check yourself

1. Two similar triangles have areas $16$ square cm and $25$ square cm. In what ratio are their sides?

2. Is a triangle with sides $11$ cm, $60$ cm and $61$ cm right-angled?