1. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle?

(a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

2. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

3. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

4. What is the ratio of the area of an equilateral triangle to the area of its circumcircle?

5. G is the centroid of equilateral triangle ABC; if AB = 10 cm, then what is the length of AG?

(a) \(10\sqrt3\) cm (b) \(\cfrac{10}{3}\) cm (c) \(10\) cm (d) \(\cfrac{10\sqrt3}{3}\) cm

6. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

7. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?

(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

8. The sum of the length, width, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the rectangular box?

(a) 360 square cm (b) 221 square cm (c) 351quare cm (d) 256 quare cm

9. If the area of the incircle of an equilateral triangle is 49 ???? cm 2 , what is the perimeter of the triangle?

(a) 21\(\sqrt3\) (b) 7\(\sqrt3\) (c) 14\(\sqrt3\) (d) 42\(\sqrt3\)

10. In triangle △ABC, if ∠ABC = 90°, AB = 5 cm, and BC = 12 cm, then what is the radius of its circumcircle?

11. The sum of the length, breadth, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the box?

12. ABC is a right-angled triangle with hypotenuse BC. From point A, a perpendicular AD is drawn to BC. If BD = 4 cm and DC = 5 cm, then what is the length of AB?

13. If each side of an equilateral triangle is 4 cm, then the measurement of the triangle's height is —

(a) \(4\sqrt{3}\) cm (b) \(16\sqrt{3}\) cm (c) \(8\sqrt{3}\) cm (d) \(2\sqrt{3}\) cm

14. "The differences between the semi-perimeter of a triangle and the lengths of its sides are respectively 4 cm, 7 cm, and 5 cm. The area of the triangle is –"

(a) \(20\sqrt{7}\) square cm (b) \(10\sqrt{14}\) square cm (c) \(20\sqrt{14}\) square cm (d) 140 square cm

15. 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?

16. In \(\triangle\)ABC, a line parallel to side BC intersects AB and AC at points P and Q, respectively. If AP = 4 cm, QC = 9 cm, and PB = AQ, then what is the length of PB?

17. In \(\triangle\)ABC, \(\angle\)ABC = 90°, and BD \(\bot\) AC. If BD = 6 cm and AD = 4 cm, then what is the length of CD?

18. “The perimeters of two similar triangles are 24 cm and 16 cm respectively. If one side of the second triangle is 6 cm, what will be the length of the corresponding side of the first triangle.”

19. In triangle \( \triangle ABC \), if \( \angle ABC = 90^\circ \), \( AB = 6 \) cm, and \( BC = 8 \) cm, then what is the length of the circumradius of triangle \( \triangle ABC \)?

20. What is the height of an equilateral triangle ABC if each side is of length 2a units?

21. In triangle ABC, ∠B = 90° and BD ⊥ AC; If AB = 30 cm, BD = 24 cm, and AD = 18 cm, then what is the length of BC?

22. If the perimeter of a rhombus is 40 cm and one of its diagonals is 12 cm, what is the area of the rhombus?

23. In right-angled triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \) and BD is perpendicular to AC. If BD = 8 cm and AD = 4 cm, what is the length of CD?

24. In the circle with center \( O \), if \( AOB \) is a diameter and point \( C \) is on the circle such that \( AC = 3 \) cm and \( BC = 4 \) cm, what is the length of \( AB \)?

(a) 3 cm (b) 4 cm (c) 5 cm (d) 8 cm

25. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?

(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

26. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

27. AB and CD are two parallel straight lines. AD and BC intersect each other at point O. If OA = 2 cm, OB = 3 cm, and OD = 4 cm, then what is the length of OC?

(a) 6 cm (b) 4 cm (c) 4.8 cm (d) 4.2 cm

28. What is the radius of the circumcircle of a triangle whose sides are 20 cm, 21 cm, and 29 cm?

(a) 14\(\frac{1}{2}\) cm (b) 14 cm (c) 10 cm (d) 11\(\frac{1}{2}\) cm

29. The three sides of a triangle are 4 cm, 5 cm, and 7 cm respectively. Therefore, the triangle is –

(a) acute-angled (b) obtuse-angled (c) right-angled (d) None of the above

30. What is the radius of the incircle of an equilateral triangle whose each side is \(2a\)?

(a) \(\sqrt{3a}\) cm (b) \(\cfrac{a}{\sqrt3}\) cm (c) \(\cfrac{a}{\sqrt2}\) cm (d) \(\sqrt{2a}\) cm