Similar Triangles

Similar Triangles

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: $$ $\angle A=\angle D$, $\angle B=\angle E$, $\angle C=\angle F$ We say that $\triangle ABC ~\sim~\triangle DEF$.

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

Construct a point along a segment at a given ratio

Using Technology

1. Draw a point off the line segment $\overline{AB}$. Label this as point $C$.
2. Draw the ray $\overrightarrow{AC}$step.
3. Use the circle tool circle centered at $A$. Label this as point $P_1$.
4. Use the compass tool to draw a circle of radius $\overline{AP_1}$, centered at $P_1$. Label the intersection of the circle and the ray as $P_2$ .
5. Repeat step 4 to mark out equally spaced points until you reach $P_5$.
6. Draw the segment $\overline{P_5 B}$
7. Construct a line parallel to $\overline{P_5 B}$ , and through the point $P_4$. Label the intersection of the line and the segment $\overline{AB}$ as $Q_4$.
8. Repeat step 7 for all of the points $P_i$
9. The points $Q_1$ to $Q_4$ subdivide $\overline{AB}$ into 5 congruent segments.
10. Relabel $Q_2$ as point $D$.