1. In triangle ABC, if AB = (2a−1) cm, AC = \(2\sqrt{2}a\) cm, and BC = (2a+1) cm, then write the value of ∠BAC.

2. In triangle △ABC, if AB = \((2a - 1)\) cm, AC = \(2\sqrt{2a}\) cm, and BC = \((2a + 1)\) cm, then find the measure of ∠BAC.

3. In \(\triangle\)ABC, if AB \(= (2p-1)\) cm, AC \(= 2\sqrt2p\) cm, and BC \(= (2p+1)\) cm, then the value of \(\angle\)BAC is...? Let me know if you need further assistance!

4. In the figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then what is the value of x?

5. In the adjacent circle centered at O, OP ⊥ AB; if AB = 6 cm and PC = 2 cm, then calculate and write the length of the radius of the circle.

6. In the adjacent figure, DE || BC in triangle \( \triangle ABC \). If \( AD = 5 \) cm, \( DB = 6 \) cm, and \( AE = 7.5 \) cm, then calculate and write the length of \( AC \).

7. Given that \(\triangle ABC\) has \(\angle ABC = 90^\circ\) and \(BD \perp AC\); if \(BD = 6\) cm and \(AD = 4\) cm, then calculate and write the length of \(CD\).

8. In \(\triangle ABC\), \(\angle ABC = 90^\circ\) and \(BD \perp AC\); if \(AB = 6\) cm, \(BD = 3\) cm, and \(CD = 5.4\) cm, then calculate and write the length of side \(BC\).

9. In right-angled triangle ABC, ∠B is the right angle. If AB = \(8\sqrt{3}\) cm and BC = 8 cm, then calculate the values of ∠ACB and ∠BAC.

10. From point A of triangle ABC, a perpendicular AD is drawn to side BC, meeting at point D. If BD = 8 cm, DC = 2 cm, and AD = 4 cm, then write the measure of ∠BAC.

11. In triangle ∆ABC, \(\angle ABC = 90^\circ\) and BD ⊥ AC. If BD = 8 cm and AD = 5 cm, then calculate and write the length of CD.

12. ABC is a right-angled triangle with \(\angle B\) being the right angle and BD ⟂ AC. If AD = 4 cm and CD = 16 cm, then calculate and write the lengths of BD and AB.

13. In the adjacent figure, ∠ACB = ∠BAD and AD is perpendicular to BC; if AC = 15 cm, AB = 20 cm, and BC = 25 cm, then write the length of AD.

14. In the adjacent figure, ∠ABC = 90° and BD is perpendicular to AC; if AB = 30 cm, BD = 24 cm, and AD = 18 cm, then write the length of BC.

15. In the adjacent figure, ∠ABC = 90° and BD is perpendicular to AC; if BD = 8 cm and AD = 4 cm, then write the length of CD.

16. In the adjacent figure, within triangle \(ABC\), if \(DE \parallel PQ \parallel BC\) and \(AD = 3\) cm, \(DP = x\) cm, \(PB = 4\) cm, \(AE = 4\) cm, \(EQ = 5\) cm, and \(QC = y\) cm, then determine the values of \(x\) and \(y\).

17. In the adjacent figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then find the value of x.