1. **State the problem:** We have a rectangular pyramid with base ABCD where AB = 7 cm and BC = 9 cm. The apex is T, and all edges TA, TB, TC, and TD are 12 cm. M is the midpoint of AD. We need to find the angle $\angle TAC$ to 1 decimal place. 2. **Understand the geometry:** The base ABCD is a rectangle with sides 7 cm and 9 cm. The diagonal AC can be found using the Pythagorean theorem: $$AC = \sqrt{7^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4018\, \text{cm}$$ 3. **Find coordinates for points:** To analyze the angle, place the rectangle in the coordinate plane for convenience. Let: - A = (0,0,0) - B = (7,0,0) - C = (7,9,0) - D = (0,9,0) Since M is midpoint of AD: $$M = \left(\frac{0+0}{2}, \frac{0+9}{2}, 0\right) = (0,4.5,0)$$ 4. **Find the height TM:** Since TA = TB = TC = TD = 12 cm, and base points are on the xy-plane (z=0), T lies vertically above M at height h. Calculate TM using Pythagoras in triangle TDA: Distance DA = 9 cm (side AD) Since M is midpoint of AD, TM is vertical height. Use TA = 12 cm and AM = half of AD = 4.5 cm. Distance AM = 4.5 cm Using right triangle TAM: $$TA^2 = TM^2 + AM^2 \Rightarrow TM = \sqrt{TA^2 - AM^2} = \sqrt{12^2 - 4.5^2} = \sqrt{144 - 20.25} = \sqrt{123.75} \approx 11.125\, \text{cm}$$ So, T = (0,4.5,11.125) 5. **Find vectors TA and AC:** $$\vec{TA} = A - T = (0-0, 0-4.5, 0-11.125) = (0, -4.5, -11.125)$$ $$\vec{AC} = C - A = (7-0, 9-0, 0-0) = (7, 9, 0)$$ 6. **Calculate the angle $\angle TAC$:** The angle between vectors $\vec{TA}$ and $\vec{AC}$ is given by: $$\cos \theta = \frac{\vec{TA} \cdot \vec{AC}}{|\vec{TA}| |\vec{AC}|}$$ Calculate dot product: $$\vec{TA} \cdot \vec{AC} = 0 \times 7 + (-4.5) \times 9 + (-11.125) \times 0 = -40.5$$ Calculate magnitudes: $$|\vec{TA}| = \sqrt{0^2 + (-4.5)^2 + (-11.125)^2} = \sqrt{0 + 20.25 + 123.7656} = \sqrt{144.0156} \approx 12$$ $$|\vec{AC}| = \sqrt{7^2 + 9^2 + 0^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4018$$ 7. **Compute cosine and angle:** $$\cos \theta = \frac{-40.5}{12 \times 11.4018} = \frac{-40.5}{136.8216} \approx -0.2959$$ $$\theta = \cos^{-1}(-0.2959) \approx 107.2^\circ$$ 8. **Final answer:** The angle $\angle TAC$ is approximately **107.2 degrees** to 1 decimal place.