1. **Problem statement:** We have a right triangle with a vertical height labeled $h$, a base segment labeled $x$, and given side lengths: the left vertical side is 65, the top horizontal segment is 144, and the hypotenuse is 156. We need to find the values of $h$ and $x$. 2. **Understanding the problem:** The triangle is right-angled, so we can use the Pythagorean theorem: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Given:** The large triangle has legs 65 and 144, and hypotenuse 156. Check if this satisfies the Pythagorean theorem: $$65^2 + 144^2 = 4225 + 20736 = 24961$$ $$156^2 = 24336$$ Since $24961 \neq 24336$, the triangle with sides 65, 144, and 156 is not a right triangle. However, the problem states it is right-angled, so likely the smaller triangle inside shares height $h$ and base $x$. 4. **Using similar triangles:** The smaller right triangle inside shares height $h$ and base $x$. Since the triangles are similar, the ratios of corresponding sides are equal: $$\frac{h}{65} = \frac{x}{144} = \frac{\text{hypotenuse of small triangle}}{156}$$ 5. **Find $h$ and $x$:** From the problem, the hypotenuse of the large triangle is 156, and the smaller triangle shares height $h$ and base $x$. Using the Pythagorean theorem for the large triangle: $$65^2 + 144^2 = 156^2$$ Check: $$4225 + 20736 = 24336$$ $$24961 \neq 24336$$ This suggests a mistake in the problem statement or measurements. 6. **Assuming the smaller triangle is right-angled with height $h$ and base $x$, and the large triangle has height 65 and base 144:** Using similarity: $$\frac{h}{65} = \frac{x}{144}$$ 7. **Given your answer $h=60$ and $x=144$, check if this satisfies the ratio:** $$\frac{60}{65} = 0.9231$$ $$\frac{144}{144} = 1$$ They are not equal, so the answer $x=144$ is likely incorrect. 8. **Calculate $x$ using the ratio:** $$x = \frac{144}{65} \times h$$ If $h=60$, then $$x = \frac{144}{65} \times 60 = \frac{144 \times 60}{65} = \frac{8640}{65} = 132.92$$ 9. **Final answers:** $$h = 60$$ $$x \approx 132.92$$ Thus, the correct value of $x$ is approximately 132.92, not 144.