1. **Problem statement:** We have a circle with center $O$ and radius $OA = 7.5$ cm. The radius $OA$ is perpendicular to chord $AB$. Point $D$ lies on the circle such that $\angle D = 90^\circ$. From an external point $E$, two segments $EA$ and $ED$ form equal angles of $35^\circ$ each at $E$. Points $B$ and $C$ are also on the circle. We want to analyze the geometric relationships and find relevant lengths or angles based on this setup. 2. **Key formulas and rules:** - The radius perpendicular to a chord bisects the chord. - In a circle, an inscribed angle subtending a diameter is a right angle ($90^\circ$). - The sum of angles around point $E$ involving $EA$ and $ED$ is $70^\circ$ (two $35^\circ$ angles). 3. **Step 1: Use the perpendicular radius to chord property** Since $OA$ is perpendicular to chord $AB$, $OA$ bisects $AB$. Let $M$ be the midpoint of $AB$. Then $AM = MB = x$. 4. **Step 2: Calculate length $AB$ using right triangle $OAM$** Since $OA = 7.5$ cm and $OM$ is the distance from center $O$ to midpoint $M$ on chord $AB$, by Pythagoras: $$OA^2 = OM^2 + AM^2$$ We know $OA = 7.5$ cm, so: $$7.5^2 = OM^2 + x^2$$ 5. **Step 3: Use the right angle at $D$** Since $\angle D = 90^\circ$ and $D$ lies on the circle, $AD$ subtends a diameter. This means $AD$ is a diameter of the circle. Therefore, length $AD = 2 \times OA = 2 \times 7.5 = 15$ cm. 6. **Step 4: Analyze angles at $E$** At point $E$, $\angle E = 35^\circ$ for both $EA$ and $ED$. This suggests $E$ lies on the angle bisector or forms an isosceles triangle with $A$ and $D$. 7. **Summary:** - Radius $OA = 7.5$ cm - Chord $AB$ is bisected by $OA$ - $AD$ is diameter $= 15$ cm - Angles at $E$ are $35^\circ$ each Without additional numeric data or specific questions (e.g., find length $AB$, $EB$, or $EC$), this is the geometric analysis based on the given information.