1. **Problem statement:** Find the unknown sides in the similar triangles from the first diagram, starting with TV. 2. **Step 1: Understand the problem and given data.** - Segments on the base: RK = 2 cm, KC = 2.5 cm, CV = 3 cm. - Vertical height TR = 4 cm. - Hypotenuse TV is unknown. - Points B and H lie on CV and TR respectively. 3. **Step 2: Use similarity of triangles.** - Since triangles are similar, corresponding sides are proportional. - The total base length RC = RK + KC + CV = 2 + 2.5 + 3 = 7.5 cm. 4. **Step 3: Find TV using Pythagoras or similarity.** - Given the vertical height TR = 4 cm and base RC = 7.5 cm, TV is the hypotenuse of right triangle TRV. - Use Pythagoras theorem: $$TV = \sqrt{TR^2 + RC^2} = \sqrt{4^2 + 7.5^2} = \sqrt{16 + 56.25} = \sqrt{72.25} = 8.5\,cm$$ 5. **Step 4: Find HK.** - HK corresponds to a segment proportional to KC = 2.5 cm. - Using similarity ratio: $$\frac{HK}{KC} = \frac{TR}{TV} = \frac{4}{8.5}$$ - Solve for HK: $$HK = 2.5 \times \frac{4}{8.5} = \frac{10}{8.5} = 1.176\,cm$$ 6. **Step 5: Find BC.** - BC corresponds to segment CV = 3 cm. - Using similarity ratio: $$\frac{BC}{CV} = \frac{TR}{TV} = \frac{4}{8.5}$$ - Solve for BC: $$BC = 3 \times \frac{4}{8.5} = \frac{12}{8.5} = 1.412\,cm$$ 7. **Step 6: Find VH.** - VH is vertical segment corresponding to TR minus HK: $$VH = TR - HK = 4 - 1.176 = 2.824\,cm$$ 8. **Step 7: Find VB.** - VB corresponds to CV minus BC: $$VB = CV - BC = 3 - 1.412 = 1.588\,cm$$ **Final answers:** - $$TV = 8.5\,cm$$ - $$HK \approx 1.18\,cm$$ - $$BC \approx 1.41\,cm$$ - $$VH \approx 2.82\,cm$$ - $$VB \approx 1.59\,cm$$