1. **Problem statement:** Ram starts at entrance E at the northernmost point of a circle with radius 50m. He walks 1/4 of the circle clockwise, then cuts across the circle to point P on the opposite side such that EP passes through the center. From P, he walks tangentially 30m to point T. We need to find the direction of T from E. 2. **Understanding the problem:** - The circle radius $r=50$ m. - 1/4 of the circle corresponds to an arc length of $\frac{1}{4} \times 2\pi r = \frac{1}{4} \times 2\pi \times 50 = 25\pi$ m. - Walking 1/4 circle clockwise from the northmost point E means moving 90° clockwise along the circle. 3. **Coordinates setup:** Place the center of the circle at origin $O(0,0)$. - Entrance $E$ is at the northmost point: $E(0,50)$. - After walking 1/4 circle clockwise, Ram reaches point $Q$ at 90° clockwise from north, which is the eastmost point: $Q(50,0)$. 4. **Point P:** P is on the opposite side of the circle such that $EP$ passes through the center. Since $E$ is at $(0,50)$, the opposite point $P$ is at $(0,-50)$. 5. **From P, Ram walks tangentially 30m to T:** - The tangent at $P(0,-50)$ is horizontal because the radius vector $OP$ is vertical. - The tangent direction at $P$ is along the x-axis. - Ram can walk either left or right along the tangent. 6. **Finding T coordinates:** - Walking 30m tangentially from $P(0,-50)$: - To the right: $T(30,-50)$ - To the left: $T(-30,-50)$ 7. **Direction of T from E:** - Vector $\overrightarrow{ET} = T - E$. - For $T(30,-50)$: $\overrightarrow{ET} = (30 - 0, -50 - 50) = (30, -100)$. - For $T(-30,-50)$: $\overrightarrow{ET} = (-30 - 0, -50 - 50) = (-30, -100)$. 8. **Calculate angles:** - Angle $\theta = \arctan\left(\frac{\Delta y}{\Delta x}\right)$ from east axis. - For $(30,-100)$: $\theta = \arctan\left(\frac{-100}{30}\right) \approx -73.3^\circ$ (measured clockwise from east). - For $(-30,-100)$: $\theta = \arctan\left(\frac{-100}{-30}\right) = \arctan(3.33) \approx 73.3^\circ$ but in the third quadrant, so angle from east is $180 - 73.3 = 106.7^\circ$. 9. **Convert to compass directions:** - $-73.3^\circ$ from east is $73.3^\circ$ south of east (southeast direction). - $106.7^\circ$ from east is $16.7^\circ$ south of west (southwest direction). 10. **Which direction is correct?** - Ram walked clockwise along the circle, so the tangent direction at P is towards the east (right) side. - Therefore, $T$ is at $(30,-50)$. **Final answer:** Point $T$ is approximately in the southeast direction from $E$.