1. **Problem 1: Find the missing numbers given median and range** We have four number cards: 1, 5, ?, ? Given: - Median = 6 - Range = 14 2. **Step 1: Understand median and range** - Median of 4 numbers is the average of the 2nd and 3rd numbers when sorted. - Range = largest number - smallest number 3. **Step 2: Sort known numbers and unknowns** Known numbers: 1, 5 Unknown numbers: $x$, $y$ Sorted order: 1, 5, $x$, $y$ or 1, $x$, 5, $y$ etc. We must find the correct order. 4. **Step 3: Use median = 6** Median = average of 2nd and 3rd numbers = 6 So, $\frac{a_2 + a_3}{2} = 6 \Rightarrow a_2 + a_3 = 12$ 5. **Step 4: Use range = 14** Range = largest - smallest = 14 Smallest number is 1 (given) Largest number = 1 + 14 = 15 6. **Step 5: Determine order** Since 1 is smallest and 15 is largest, the four numbers are 1, ?, ?, 15 Median is average of 2nd and 3rd numbers = 6 So, $a_2 + a_3 = 12$ 7. **Step 6: Known numbers are 1 and 5, but 5 is less than 6, so 5 must be one of the middle numbers** Try order: 1, 5, $x$, 15 Median = $\frac{5 + x}{2} = 6 \Rightarrow 5 + x = 12 \Rightarrow x = 7$ 8. **Step 7: Final numbers are 1, 5, 7, 15** --- 9. **Problem 2: Find length $d$ in the right triangle** Given: - Base split into 6.2 cm and 3.8 cm segments - Hypotenuse length $d$ - Angle $\theta$ opposite height 10. **Step 1: Write two equations involving $\cos \theta$ and $d$** - The total base length = 6.2 + 3.8 = 10 cm - Using cosine definition: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ 11. **Step 2: Express adjacent side in terms of $d$ and $\theta$** - Adjacent side = 10 cm - So, $\cos \theta = \frac{10}{d} \Rightarrow d = \frac{10}{\cos \theta}$ 12. **Step 3: Use sine definition for height** - Height = $h$ - $\sin \theta = \frac{h}{d} \Rightarrow h = d \sin \theta$ 13. **Step 4: Use Pythagoras theorem** - Height $h$ is also the vertical side - Base = 10 cm - Hypotenuse = $d$ - So, $d^2 = h^2 + 10^2$ 14. **Step 5: Substitute $h = d \sin \theta$** $$d^2 = (d \sin \theta)^2 + 10^2 = d^2 \sin^2 \theta + 100$$ 15. **Step 6: Rearrange to solve for $d$** $$d^2 - d^2 \sin^2 \theta = 100$$ $$d^2 (1 - \sin^2 \theta) = 100$$ $$d^2 \cos^2 \theta = 100$$ $$d = \frac{10}{\cos \theta}$$ 16. **Step 7: Calculate $d$ using given segments** - Since $\cos \theta = \frac{10}{d}$, and $d$ is hypotenuse, use Pythagoras with segments 6.2 and 3.8 - Hypotenuse $d = \sqrt{6.2^2 + 3.8^2} = \sqrt{38.44 + 14.44} = \sqrt{52.88} \approx 7.27$ cm 17. **Step 8: Final answer** Length $d \approx 7.27$ cm (to 2 decimal places)