Triangles

The One Thing

Triangle similarity establishes equal corresponding angles and proportional corresponding sides; Pythagoras’ theorem relates the sides of a right-angled triangle. Correctly identifying correspondence, parallel-line relationships and the hypotenuse is essential when forming proportions or calculating lengths.

Definitions and Results

• Similar triangles: Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.
• Corresponding sides: Sides that lie opposite equal corresponding angles in two triangles.
• Corresponding angles: Angles in two figures that occupy matching positions and have equal measures.
• AAA similarity criterion: If the three angles of one triangle are respectively equal to the three angles of another triangle, the triangles are similar.
• AA similarity criterion: If two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar because the third angles are also equal. This follows from the fact that the angles in a triangle sum to $180^{\circ }$.
• SSS similarity criterion: If the three corresponding sides of two triangles are proportional, the triangles are similar.
• SAS similarity criterion: If one pair of corresponding angles is equal and the sides including those angles are proportional, the triangles are similar.
• Scale factor: The ratio of any corresponding side of one similar triangle to the corresponding side of the other triangle.
• Similarity notation: If triangle $ABC$ is similar to triangle $PQR$, write

$$△ABC\sim △PQR.$$ The order gives the correspondence $A\leftrightarrow P$, $B\leftrightarrow Q$, and $C\leftrightarrow R$.

• Corresponding sides in similar triangles:

$$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{RP}.$$

• Corresponding angles in similar triangles:

$$\angle A=\angle P,\angle B=\angle Q,\angle C=\angle R.$$

• Areas of similar triangles:

$$\frac{\mathrm{ar}(△ABC)}{\mathrm{ar}(△PQR)}={\left(\frac{AB}{PQ}\right)}^2.$$ Thus, the ratio of the areas equals the square of the ratio of corresponding sides.

• Perimeters of similar triangles: The ratio of the perimeters equals the ratio of corresponding sides.
• Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those two sides in the same ratio. In $△ABC$, if $DE\parallel BC$, with $D$ on $AB$ and $E$ on $AC$, then

$$\frac{AD}{DB}=\frac{AE}{EC}.$$

• Converse of the Basic Proportionality Theorem: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
• Pythagoras theorem: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If $\angle B=90^{\circ }$ in $△ABC$, then $AC$ is the hypotenuse and

$$A{C}^2=A{B}^2+B{C}^2.$$

• Hypotenuse: The side opposite the right angle in a right-angled triangle; it is always the longest side.
• Missing leg of a right triangle:

$${\text{leg}}^2={\text{hypotenuse}}^2-{\text{other leg}}^2.$$

• Finding the hypotenuse:

$$\text{hypotenuse}=\sqrt{{\text{first leg}}^2+{\text{second leg}}^2}.$$

• Converse of Pythagoras theorem: If the square of one side of a triangle equals the sum of the squares of the other two sides, the angle opposite that side is a right angle.
• Classification using the longest side: For a triangle with sides $a$, $b$ and $c$, where $c$ is the longest side:

- $c^2=a^2+b^2$ indicates a right angle opposite $c$; - $c^2<a^2+b^2$ indicates an acute triangle; - $c^2>a^2+b^2$ indicates an obtuse triangle.

• Applicability: Pythagoras theorem applies only to right-angled triangles, whereas similarity criteria apply to any triangles.
• Units: Length units must be the same before using ratios or applying Pythagoras theorem.

Worked Methods

1. Proving similarity using angle and side criteria

• Identify the vertices, angles and sides that correspond.
• For AA similarity, show that two corresponding angles are equal. The third pair is then equal because the angles of a triangle sum to $180^{\circ }$.
• For AAA similarity, show that all three corresponding angles are equal.
• For SSS similarity, compare all three pairs of corresponding sides and verify that their ratios are equal.
• For SAS similarity, show that one pair of corresponding angles is equal and that the two sides including those angles are proportional.
• State the similarity using the correct vertex order. For example:

$$△ABC\sim △PQR.$$

• Use the correspondence to write:

$$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{RP}.$$

Parallel lines can establish similarity because they produce equal corresponding angles. A line parallel to one side of a triangle therefore creates a smaller triangle similar to the original triangle.

2. Applying similarity to areas and perimeters

• Establish that the two triangles are similar.
• Identify a pair of corresponding sides, such as $AB$ and $PQ$.
• For areas, square the side ratio:

$$\frac{\mathrm{ar}(△ABC)}{\mathrm{ar}(△PQR)}={\left(\frac{AB}{PQ}\right)}^2.$$

• For perimeters, use the side ratio without squaring:

$$\frac{\text{perimeter of }△ABC}{\text{perimeter of }△PQR}=\frac{AB}{PQ}.$$

3. Applying the Basic Proportionality Theorem

In $△ABC$, suppose $D$ lies on $AB$, $E$ lies on $AC$, and $DE\parallel BC$.

• Identify the parallel lines $DE$ and $BC$.
• Use the Basic Proportionality Theorem:

$$\frac{AD}{DB}=\frac{AE}{EC}.$$

• If required, use the similarity created by the parallel line:

$$△ADE\sim △ABC.$$

• Apply the resulting ratios:

$$\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}.$$

4. Using the converse of the Basic Proportionality Theorem

• Check whether a line divides two sides of a triangle in the same ratio.
• For the configuration involving $D$ on $AB$ and $E$ on $AC$, verify:

$$\frac{AD}{DB}=\frac{AE}{EC}.$$

• Conclude that the dividing line is parallel to the third side:

$$DE\parallel BC.$$

5. Finding an unknown side using Pythagoras theorem

• Confirm that the triangle is right-angled.
• Identify the hypotenuse as the side opposite the right angle.
• Ensure that all length units are the same.
• If $\angle B=90^{\circ }$ in $△ABC$, use:

$$A{C}^2=A{B}^2+B{C}^2.$$

• To find a missing leg, rearrange:

$${\text{leg}}^2={\text{hypotenuse}}^2-{\text{other leg}}^2.$$

• To find the hypotenuse, use:

$$\text{hypotenuse}=\sqrt{{\text{first leg}}^2+{\text{second leg}}^2}.$$

6. Classifying a triangle using the converse of Pythagoras theorem

• Identify the longest side and call it $c$.
• Let the other sides be $a$ and $b$.
• Compare $c^2$ with $a^2+b^2$:

- If $c^2=a^2+b^2$, the angle opposite $c$ is a right angle. - If $c^2<a^2+b^2$, the triangle is acute. - If $c^2>a^2+b^2$, the triangle is obtuse.

Where It Goes Wrong

• Corresponding vertices are mismatched. The order in $△ABC\sim △PQR$ must preserve $A\leftrightarrow P$, $B\leftrightarrow Q$, and $C\leftrightarrow R$.
• Proportions compare non-corresponding sides. Ratios must use matching sides, such as

$$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{RP}.$$

• The condition $DE\parallel BC$ is omitted when applying the Basic Proportionality Theorem.
• The converse is used without first verifying that the two side divisions are in the same ratio.
• Pythagoras theorem is applied to a triangle that is not right-angled. Similarity criteria apply to any triangles, but Pythagoras theorem does not.
• The longest side is not identified as the hypotenuse, or the length units are not made consistent before calculation.

What Gets Asked

• Determine whether two triangles are similar using AAA, AA, SSS or SAS.
• Write a correct similarity statement and identify corresponding vertices, angles and sides.
• Calculate an unknown side using proportional corresponding sides.
• Find the scale factor between similar triangles.
• Calculate the ratio of the areas or perimeters of similar triangles.
• Apply the Basic Proportionality Theorem in the configuration $DE\parallel BC$ in $△ABC$.
• Use the converse of the Basic Proportionality Theorem to prove that a line is parallel to the third side.
• Prove that a smaller triangle formed by a line parallel to one side is similar to the original triangle.
• Find a missing leg or the hypotenuse using Pythagoras theorem.
• Use the converse of Pythagoras theorem to determine whether a triangle is right-angled, acute or obtuse.
• Solve geometrical problems involving unknown lengths, heights, distances and indirect measurement.