1. **Problem Statement:** Find the angle of $\angle HAM$ and the length of segment $MT$ in triangle $HAM$ where $AM = 26$ cm, $HM = 46$ cm, and $\angle M$ is split into two equal angles of $24.5^\circ$ each. 2. **Identify the triangle type and applicable formulas:** Since $HAM$ is a triangle with given sides and angles, and $\angle M$ is split into two equal parts, we can use the Law of Cosines and Law of Sines. 3. **Calculate $\angle HAM$:** We know $AM = 26$ cm, $HM = 46$ cm, and $\angle M = 24.5^\circ + 24.5^\circ = 49^\circ$. Using Law of Cosines to find $HA$: $$HA^2 = HM^2 + AM^2 - 2 \times HM \times AM \times \cos(\angle M)$$ $$HA^2 = 46^2 + 26^2 - 2 \times 46 \times 26 \times \cos(49^\circ)$$ Calculate each term: $$46^2 = 2116$$ $$26^2 = 676$$ $$2 \times 46 \times 26 = 2392$$ $$\cos(49^\circ) \approx 0.6561$$ So, $$HA^2 = 2116 + 676 - 2392 \times 0.6561$$ $$HA^2 = 2792 - 1570.5 = 1221.5$$ $$HA = \sqrt{1221.5} \approx 34.95\text{ cm}$$ 4. **Find $\angle HAM$ using Law of Sines:** Law of Sines states: $$\frac{\sin(\angle HAM)}{HM} = \frac{\sin(\angle M)}{HA}$$ Rearranged: $$\sin(\angle HAM) = \frac{HM}{HA} \times \sin(\angle M)$$ Calculate: $$\sin(\angle HAM) = \frac{46}{34.95} \times \sin(49^\circ)$$ $$\sin(49^\circ) \approx 0.7547$$ $$\sin(\angle HAM) = 1.316 \times 0.7547 = 0.993$$ So, $$\angle HAM = \arcsin(0.993) \approx 83.7^\circ$$ 5. **Find length $MT$:** Point $T$ lies on $HA$ such that $\angle M$ is split into two equal angles of $24.5^\circ$. This means $T$ is the foot of the angle bisector from $M$ to $HA$. Using the Angle Bisector Theorem: The angle bisector divides the opposite side into segments proportional to the adjacent sides: $$\frac{AT}{TH} = \frac{AM}{HM} = \frac{26}{46} = \frac{13}{23}$$ Since $HA = 34.95$ cm, let $AT = x$, then $TH = 34.95 - x$. Set up the proportion: $$\frac{x}{34.95 - x} = \frac{13}{23}$$ Cross multiply: $$23x = 13(34.95 - x)$$ $$23x = 454.35 - 13x$$ $$23x + 13x = 454.35$$ $$36x = 454.35$$ $$x = \frac{454.35}{36} \approx 12.62\text{ cm}$$ So, $AT = 12.62$ cm and $TH = 34.95 - 12.62 = 22.33$ cm. 6. **Calculate $MT$ using Law of Sines in triangle $MTM$:** In triangle $MTM$, $\angle M$ is split into two $24.5^\circ$ angles. Using the right triangle formed by the bisector, $MT$ is the length from $M$ perpendicular to $HA$. Using the formula for the length of the angle bisector: $$MT = \frac{2 \times AM \times HM \times \cos(24.5^\circ)}{AM + HM}$$ Calculate: $$MT = \frac{2 \times 26 \times 46 \times \cos(24.5^\circ)}{26 + 46}$$ $$\cos(24.5^\circ) \approx 0.910$$ $$MT = \frac{2 \times 26 \times 46 \times 0.910}{72} = \frac{2177.92}{72} \approx 30.25\text{ cm}$$ **Final answers:** - $\angle HAM \approx 83.7^\circ$ - $MT \approx 30.25$ cm