Identifying Similar Triangles

Triangle $ABC$ has angles $50°$, $60°$, and $70°$. Triangle $DEF$ has angles $50°$, $60°$, and $70°$. Are they similar?

Compare corresponding angles: Angle A = $50°$ = Angle D Angle B = $60°$ = Angle E Angle C = $70°$ = Angle F = All angles match

Apply the AA (Angle-Angle) criterion: If two angles of one triangle equal two angles of another triangle, they are similar = AA criterion satisfied

State the conclusion: Since all corresponding angles are equal, the triangles are similar = $\triangle ABC \sim \triangle DEF$

Answer: Yes, the triangles are similar because all corresponding angles are equal.

Finding the Scale Factor

Rectangle $ABCD$ has sides of $6$ cm and $10$ cm. Rectangle $EFGH$ has sides of $9$ cm and $15$ cm. Are they similar? If so, what is the scale factor?

Check if corresponding sides are proportional: Ratio of shorter sides: $\frac{9}{6} = 1.5$ Ratio of longer sides: $\frac{15}{10} = 1.5$ = Both ratios are equal

Verify all angles are equal: All angles in a rectangle are $90°$ = Angles match

Determine the scale factor: Since all side ratios equal $1.5$, the scale factor is $1.5$ (or $\frac{3}{2}$) = Scale factor = $1.5$

Answer: Yes, the rectangles are similar with a scale factor of $1.5$.

Finding a Missing Side Using Similarity

Triangle $PQR \sim$ Triangle $STU$. In $\triangle PQR$: $PQ = 8$ cm, $QR = 12$ cm. In $\triangle STU$: $ST = 12$ cm. Find $TU$.

Identify corresponding sides: $PQ$ corresponds to $ST$ $QR$ corresponds to $TU$ = Pairs identified

Find the scale factor: Scale factor = $\frac{ST}{PQ} = \frac{12}{8} = 1.5$ = Scale factor = $1.5$

Apply the scale factor to find $TU$: $TU = QR \times 1.5 = 12 \times 1.5 = 18$ = $TU = 18$ cm

Verify using proportion: $\frac{PQ}{ST} = \frac{QR}{TU}$ $\frac{8}{12} = \frac{12}{18}$ $\frac{2}{3} = \frac{2}{3}$ ✓ = Answer verified

Answer: $TU = 18$ cm

Mistake: Confusing similar with congruent

Why: Congruent figures are exactly the same size AND shape. Similar figures only need to be the same shape - they can be different sizes.

Correct: All congruent figures are similar (with scale factor 1), but similar figures are not necessarily congruent.

Mistake: Matching sides incorrectly when comparing figures

Why: You must match sides that are in the same position relative to the angles, not just sides of the same length.

Correct: Always identify corresponding angles first, then match the sides that are opposite to those angles.

Mistake: Using the wrong scale factor direction

Why: The scale factor depends on which figure you're scaling from. Going from small to large gives a factor > 1, large to small gives a factor < 1.

Correct: Always clarify: 'Scale factor from figure A to figure B.' If $A$ is smaller, the scale factor will be greater than 1.

Scale Models in Architecture

Architects build scale models of buildings to show clients and test designs before construction.

A 1:100 scale model means every 1 cm on the model represents 100 cm (1 meter) on the actual building. A 50-meter-tall building would be 50 cm in the model.

Map Reading

Maps are similar figures to the actual terrain they represent, using a scale to convert distances.

On a map with scale 1:25000, 4 cm on the map represents 4 × 25000 = 100000 cm = 1 km of actual distance.

Q: What is the difference between similar and congruent?

A: Congruent figures are exactly the same size AND shape (scale factor = 1). Similar figures have the same shape but can be different sizes. All congruent figures are similar, but not all similar figures are congruent.

Q: Can a scale factor be less than 1?

A: Yes! When the second figure is smaller than the first, the scale factor is less than 1. For example, if figure B is half the size of figure A, the scale factor from A to B is $0.5$ or $\frac{1}{2}$.

Q: Are all squares similar to each other?

A: Yes! All squares have four right angles (90°) and four equal sides. Since the angles match and the sides are proportional (all multiplied by the same scale factor), all squares are similar to each other.