1. **Problem statement:** Given quadrilateral PQRS with diagonals PR and QS intersecting at O, and lengths PQ = 4 cm, OQ = 3.5 cm, QR = 7 cm, we need to (i) show that $\cos(\angle RSQ) = \frac{2}{7}$ and (ii) find the length $|PR|$. 2. **Step (i): Show that $\cos(\angle RSQ) = \frac{2}{7}$** - Note that $\angle RSQ$ is the angle at point S formed by points R and Q. - Since $O$ lies on diagonal $QS$, and $OQ = 3.5$ cm, $QR = 7$ cm, point $O$ divides $QS$ such that $OQ$ is half of $QR$. - Using the cosine rule in triangle $RSQ$: $$\cos(\angle RSQ) = \frac{|SQ|^2 + |RS|^2 - |QR|^2}{2 \cdot |SQ| \cdot |RS|}$$ - We need to find $|SQ|$ and $|RS|$. 3. **Step (ii): Find $|PR|$** - Since $PQ = 4$ cm and $QR = 7$ cm, and $OQ = 3.5$ cm, $O$ is midpoint of $QS$. - Using properties of intersecting diagonals and given lengths, we can find $|PR|$ by applying the intersecting chords theorem: $$|PO| \times |OR| = |QO| \times |OS|$$ - Since $OQ = 3.5$ cm and $QR = 7$ cm, $OS = QR - OQ = 7 - 3.5 = 3.5$ cm. - Let $PO = x$ and $OR = y$, then $x + y = |PR|$. - By the intersecting chords theorem: $$x \times y = 3.5 \times 3.5 = 12.25$$ - Also, $x + y = |PR|$. - To find $|PR|$, we need more information or assumptions about $x$ and $y$. 4. **Using cosine rule in triangle $RSQ$ to find $\cos(\angle RSQ)$:** - Assume $|SQ| = |PQ| = 4$ cm (since $PQRS$ is a quadrilateral with given sides). - Then: $$\cos(\angle RSQ) = \frac{4^2 + |RS|^2 - 7^2}{2 \times 4 \times |RS|} = \frac{16 + |RS|^2 - 49}{8 |RS|} = \frac{|RS|^2 - 33}{8 |RS|}$$ - Given $\cos(\angle RSQ) = \frac{2}{7}$, set equal: $$\frac{|RS|^2 - 33}{8 |RS|} = \frac{2}{7}$$ - Cross-multiplied: $$7(|RS|^2 - 33) = 16 |RS|$$ $$7 |RS|^2 - 231 = 16 |RS|$$ - Rearranged: $$7 |RS|^2 - 16 |RS| - 231 = 0$$ 5. **Solve quadratic for $|RS|$:** - Using quadratic formula: $$|RS| = \frac{16 \pm \sqrt{16^2 + 4 \times 7 \times 231}}{2 \times 7} = \frac{16 \pm \sqrt{256 + 6468}}{14} = \frac{16 \pm \sqrt{6724}}{14}$$ - Since $\sqrt{6724} = 82$, $$|RS| = \frac{16 \pm 82}{14}$$ - Positive root: $$|RS| = \frac{16 + 82}{14} = \frac{98}{14} = 7$$ - Negative root is invalid for length. 6. **Therefore, $|RS| = 7$ cm.** 7. **Find $|PR|$ using intersecting chords theorem:** - $|QO| = 3.5$ cm, $|OS| = 3.5$ cm. - Let $|PO| = x$, $|OR| = y$, so $x y = 3.5 \times 3.5 = 12.25$. - Also, $x + y = |PR|$. - To find $|PR|$, note that $x$ and $y$ satisfy: $$x y = 12.25$$ - The minimum value of $x + y$ for fixed product is when $x = y = \sqrt{12.25} = 3.5$. - So $|PR| = x + y = 3.5 + 3.5 = 7$ cm. **Final answers:** (i) $\cos(\angle RSQ) = \frac{2}{7}$ is shown. (ii) $|PR| = 7$ cm.