1. The circumcenter of triangle \(\triangle\)ABC lies outside the triangle. If the largest angle of the triangle is \(\angle\)BAC, then:
(a) \(\angle\)BAC = 90° (b) \(\angle\)BAC<90° (c) \(\angle\)BAC > 90° (d) \(\angle\)BAC = \(\angle\)ACB = \(\angle\)ABC
2. ABC is an acute-angled triangle. AP is the diameter of the circumcircle of triangle ABC. BE and CF are perpendiculars dropped to sides AC and AB respectively, and they intersect at point Q. Prove that BPCQ is a parallelogram.
3. The circumcenter of a------ triangle lies outside the triangle.
4. "If the two acute angles of a right-angled triangle are in the ratio 2:3, what are the radian measures of those two angles?
(a) \(\cfrac{π}{5},\cfrac{3π}{10}\) (b) \(\cfrac{π}{10},\cfrac{3π}{5}\) (c) \(\cfrac{π}{5},\cfrac{3π}{20}\) (d) \(\cfrac{π}{5},\cfrac{π}{15}\)
5. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?
(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)
6. If the three angles of a triangle are in the ratio 3:4:5, then the triangle will always be a right-angled triangle.
7. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.
8. Draw a right-angled triangle whose two sides adjacent to the right angle are 7 cm and 9 cm respectively. Then draw the incircle of that triangle. (Only construction marks are required.)
9. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?
10. If the three sides of a triangle are in the ratio 5 : 12 : 13, then the triangle will always be a right-angled triangle.
11. Draw a right-angled triangle whose hypotenuse is 10 cm and one of the other sides is 6.5 cm. Then, draw the incircle of this triangle. (Only construction marks are required.)
12. In a right-angled triangle, the difference between the two acute angles is \(\frac{2\pi}{5}\). Express the measures of these two angles in both radians and degrees.
13. In a right-angled triangle, the difference between the two acute angles is 30°. Express the measures of those two angles in both radians and degrees.
14. In a right-angled triangle, if one of the acute angles is 30°, determine the measure of the other acute angle in sexagesimal system.
15. Two acute-angled triangles ∆ABC and ∆PQR are similar. Their circumcenters are X and Y respectively. If BC and QR are corresponding (similar) sides, then prove that BX : QY = BC : QR.
16. 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.
17. Draw a right-angled triangle whose hypotenuse is 9 cm and one of the other sides is 5.5 cm. Then draw an incircle of the triangle. (Only the construction marks are required.)
18. Draw a right-angled triangle whose two sides adjacent to the right angle are 8 cm and 6 cm respectively, and draw an incircle of the triangle. (Only construction marks are required)
19. The radius, height, and slant height of a right circular cone always form the three sides of a right-angled triangle.
20. Draw a right-angled triangle whose two sides adjacent to the right angle are 4.5 cm and 6 cm. Then draw the incircle of that triangle. (Only construction marks are required.)
21. If the lengths of the three sides of a triangle are in the ratio 3:4:5, then the triangle is a right-angled triangle.
22. From the vertex A of triangle \( \triangle ABC \), a perpendicular AD is drawn to the side BC. If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that triangle ABC is a right-angled triangle.
23. In any right-angled triangle, the hypotenuse is the diameter of the circumcircle of the triangle.
24. If the difference between the measures of the two acute angles in a right-angled triangle is 10°, determine their circular measures.
25. If the ratio of the three sides of a triangle is 3:4:5, then the triangle will be a right-angled triangle.
26. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar, and each of these triangles is similar to the original triangle.
27. The height, radius, and slant height of a right circular cone always form the three sides of a right-angled triangle.
28. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.
29. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.
30. The height, radius, and slant height of a right circular cone always form the three sides of a right-angled triangle.
