1. **Stating the problem:** We are asked to find the height of a tree using indirect measurement. A 4-meter flagpole casts a 5-meter shadow, and a nearby tree casts a 20-meter shadow. We want to find the height of the tree. 2. **Formula and concept:** When two objects cast shadows at the same time, their heights and shadow lengths form similar triangles. The ratio of height to shadow length is the same for both. 3. **Set up the proportion:** Let the height of the tree be $h$. Then, $$\frac{4}{5} = \frac{h}{20}$$ 4. **Solve for $h$:** Cross-multiply: $$4 \times 20 = 5 \times h$$ $$80 = 5h$$ 5. **Divide both sides by 5:** $$\cancel{5} \times 16 = \cancel{5} \times h$$ $$16 = h$$ 6. **Answer:** The tree is 16 meters tall. --- **Next problem: Proving similarity of triangles using side lengths and angles.** 1. **Stating the problem:** Given two triangles $\triangle ABC$ and $\triangle DEF$, we want to determine if they are similar by comparing their sides and angles. 2. **Distance formula:** The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate side lengths for $\triangle ABC$:** - $AB = \sqrt{(4-2)^2 + (12-9)^2} = \sqrt{4 + 9} = \sqrt{13}$ (given) - $BC = \sqrt{(6-4)^2 + (9-12)^2} = \sqrt{4 + 9} = \sqrt{13}$ - $AC = \sqrt{(6-2)^2 + (9-9)^2} = \sqrt{16 + 0} = 4$ 4. **Calculate side lengths for $\triangle DEF$:** - $DE = \sqrt{(9-6)^2 + (9-3)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$ - $EF = \sqrt{(18-9)^2 + (6-9)^2} = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10}$ - $DF = \sqrt{(18-6)^2 + (6-3)^2} = \sqrt{144 + 9} = \sqrt{153}$ 5. **Compare ratios:** - $\frac{AB}{DE} = \frac{\sqrt{13}}{3\sqrt{5}}$ - $\frac{BC}{EF} = \frac{\sqrt{13}}{3\sqrt{10}}$ - $\frac{AC}{DF} = \frac{4}{\sqrt{153}}$ These ratios are not equal, so the triangles are not similar by side length proportionality. 6. **Angle measures:** Given angles for $\triangle ABC$ are approximately 40°, 80°, and 60°, and for $\triangle DEF$ the same angles 40°, 80°, and 60°. 7. **Conclusion:** Since all three angles are congruent, by the Angle-Angle-Angle (AAA) criterion, the triangles are similar. 8. **Complete the statement:** If three sides of one triangle are proportional to three sides of another triangle, then the two triangles are **similar**. 9. **Regarding congruence:** If three angles of one triangle are congruent to three angles of another triangle, the triangles are similar but not necessarily congruent (congruence requires side lengths to be equal).