1. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle?
2. If the area, perimeter, and height of an equilateral triangle are \(a\), \(s\), and \(h\) respectively, then the value of \(\dfrac{2a}{sh}\) is —
(a) 1 (b) \(\frac{1}{2}\) (c) \(\frac{1}{3}\) (d) \(\frac{1}{4}\)
3. “The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle.”
4. “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.”
5. 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.
6. If the perimeter of a rhombus is 40 cm and one of its diagonals is 12 cm, what is the area of the rhombus?
7. In triangle \(ABC\) and triangle \(DEF\), \(\angle ABC = \angle DEF\), \(\frac{AB}{DE} = \frac{BC}{EF}\), and the ratio of their areas is \(\frac{\triangle ABC}{\triangle DEF} = \frac{25}{16}\). If \(BC = 10\) cm, what is the length of \(EF\)?
8. 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
9. 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
10. 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
11. In the parallelogram ABCD, point P is the midpoint of side CD. If the area of triangle \( \triangle APD \) is 25 square centimeters, then what is the area of the parallelogram?
(a) 100 square cm (b) 75 square cm (c) 150 square cm (d) 50 square cm
12. 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
13. If the circumradius of an equilateral triangle is 4 cm, what is its inradius?
(a) 2 cm (b) 4 cm (c) 1 cm (d) 3 cm
14. If the length of the median drawn from the right-angled vertex of a right-angled triangle is 5 cm, what is the area of the triangle’s circumcircle?
(a) 78\(\frac78\(\frac{2}{7}\) square cm{2}{7}\ (b) 78\(\frac{3}{7}\) square cm (c) 78\(\frac{5}{7}\) square cm (d) 78\(\frac{4}{7}\) suare cm
15. In triangle ABC, \(\angle\)ABC is a right angle, and AB = 5 cm, BC = 12 cm. What is the radius of the circumcircle of triangle ABC?
16. 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
17. "If the length of the median drawn from the right angle vertex of a right-angled triangle is 10 cm, what is the area of the triangle's circumcircle?"
(a) \(100\pi\) (b) \(25\pi\) (c) \(50\pi\) (d) \(120\pi\)
18. In triangle ABC, E and F are the midpoints of sides AB and AC respectively. If the area of triangle AEF is 50 square cm, then what is the area of triangle ABC?
(a) 100 sq cm (b) 200 sq cm (c) 150 sq cm (d) 300 sq cm
19. 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
20. In a right-angled isosceles triangle ABC, ∠B is the right angle. The bisector of ∠BAC intersects BC at point D. If BD = 2 cm, then what is the length of CD?
21. In triangle ABC, ∠A = 90°, AB = 12 cm, AC = 5 cm, and BC = 13 cm. A perpendicular AD is drawn from point A to side BC. What is the length of AD?
22. The base of a triangle is \(16\sqrt{3}\) cm, and the two angles adjacent to the base are 30° and 60°. What is the height of the triangle?
23. In triangle △ABC, if ∠ABC = 90°, AB = 5 cm, and BC = 12 cm, then what is the radius of its circumcircle?
24. In triangle △ABC, ∠ABC = 90° and BD ⊥ AC. If AB = 5 cm and BC = 12 cm, then what is the length of BD?
25. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.
26. 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?
27. Point O is any point inside parallelogram ABCD. If \(\triangle AOB + \triangle COD = 16\) square cm, then what is the area of parallelogram ABCD?
(a) 8 square cm (b) 4 square cm (c) 32 square cm (d) 64 square cm
28. In triangle ABC, point D lies on side AC such that AD : DC = 3 : 2. If the area of triangle ABC is 40 square cm, then the area of triangle BDC is —
(a) 16 square cm (b) 24 square cm (c) 30 square cm (d) 36 square cm
29. "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
30. G is the centroid of triangle ABC; if the area of triangle GBC is 12 square cm, then the area of triangle ABC is —.
(a) 24 square cm (b) 6 square cm (c) 36 square cm (d) none of the above
