Review worksheet: geometry (grade 10)

Level 1 – Understanding and direct application (minimum standard for the Baccalaureate)

1. In the coordinate system $xOy$, the points $A(2,3)$ and $B(4,7)$ are given. Find the coordinates of the vector $\overrightarrow{AB}$ and its magnitude.

2. The vectors $\overrightarrow{u} = 3\overrightarrow{i} - 2\overrightarrow{j}$ and $\overrightarrow{v} = - \overrightarrow{i} + 4\overrightarrow{j}$ are given. Calculate the coordinates of the vector $\overrightarrow{w} = 2\overrightarrow{u} + 3\overrightarrow{v}$.

3. Find the length of segment $AB$ where $A( - 1,4)$ and $B(5, - 4)$.

4. Find the coordinates of the midpoint of segment $AB$, where $A( - 3,5)$ and $B(7,1)$.

5. Find the real number $a$ for which the vectors $\overrightarrow{u} = 2\overrightarrow{i} + 4\overrightarrow{j}$ and $\overrightarrow{v} = a\overrightarrow{i} + 6\overrightarrow{j}$ are collinear.

6. Consider the vectors $\overrightarrow{u} = 4\overrightarrow{i} + 3\overrightarrow{j}$ and $\overrightarrow{v} = a\overrightarrow{i} - 8\overrightarrow{j}$. Find $a \in \mathbb{R}$ such that $\overrightarrow{u}\bot\overrightarrow{v}$.

7. Find the slope of the line passing through the points $A(1,2)$ and $B(4,8)$.

8. Write the equation of the line passing through the point $A(2, - 1)$ with slope $m = - 3$.

9. Find the equation of the line passing through the points $A(2,3)$ and $B( - 1,0)$.

10. Write the explicit equation of the line $d:2x - 3y + 6 = 0$ and identify its slope.

11. The line $d:y = 2x - 5$ is given. Find the equation of the line $d'$ passing through $A(1,4)$ and parallel to $d$.

12. The line $d:y = - \frac{1}{3}x + 2$ is given. Find the slope of a line perpendicular to $d$.

13. Calculate the distance from the point $A(3,2)$ to the line $d:3x + 4y - 2 = 0$.

14. Find the centroid of the triangle with vertices $A(1,2)$, $B(5,0)$ and $C(3,7)$.

15. Check whether the points $A(1,1)$, $B(2,3)$ and $C(4,7)$ are collinear.

Level 2 – Average application (standard format, item I5 of the BAC)

16. Find the equation of the line passing through the point $A(3, - 2)$ and parallel to the line $x + 2y - 5 = 0$.

17. Find the equation of the line passing through $P( - 1,2)$ and perpendicular to the line $d:2x - y + 3 = 0$.

18. In the Cartesian coordinate system $xOy$, consider the points $A(1,2)$, $B(3,4)$ and $C(5,0)$. Find the length of the median from $A$ of triangle $ABC$.

19. Consider the points $A(2,1)$, $B(6,3)$ and $C(4,5)$. Write the equation of the altitude from vertex $C$ in triangle $ABC$.

20. Find the real number $m$ for which the lines $d_{1}:mx + y - 2 = 0$ and $d_{2}:4x + my + 1 = 0$ are parallel.

21. Find the real number $a$ for which the lines $d_{1}:(a + 1)x + 3y - 1 = 0$ and $d_{2}:2x - y + 4 = 0$ are perpendicular.

22. In the coordinate system $xOy$, the points $A(1,3)$, $B(5,1)$ and $C(3,7)$ are given. Calculate the area of triangle $ABC$.

23. Find the coordinates of the point $D$ such that the quadrilateral $ABCD$ is a parallelogram, knowing that $A(1,1)$, $B(4,2)$ and $C(5,5)$.

24. Let the vectors $\overrightarrow{u} = 2\overrightarrow{i} + (m - 1)\overrightarrow{j}$ and $\overrightarrow{v} = (m + 1)\overrightarrow{i} + 4\overrightarrow{j}$. Find $m > 0$ such that $|\overrightarrow{u}| = |\overrightarrow{v}|$.

25. The points $A( - 2,1)$ and $B(4,5)$ are given. Find the equation of the perpendicular bisector of segment $\lbrack AB\rbrack$.

26. Find the coordinates of the reflection of the point $A(3,4)$ with respect to the point $B(1,1)$.

27. Find the slope of the line $(\alpha + 1)x + (\alpha - 1)y + 3 = 0$, where $\alpha \in \mathbb{R} \smallsetminus \left\{ 1 \right\}$.

28. Show that the triangle with vertices $A(0,0)$, $B(3,1)$ and $C( - 1,3)$ is right and isosceles.

29. The vector $\overrightarrow{u} = 5\overrightarrow{i} - 12\overrightarrow{j}$ is given. Find a vector of length 1 (a unit vector) with the same direction and sense as $\overrightarrow{u}$.

30. Find the angle formed by the vectors $\overrightarrow{u} = \overrightarrow{i} + \sqrt{3}\overrightarrow{j}$ and $\overrightarrow{v} = \sqrt{3}\overrightarrow{i} + \overrightarrow{j}$.

31. Find $a \in \mathbb{R}$ such that the point $P(1,a)$ is at distance $2$ from the line $d:3x - 4y + 1 = 0$.

32. The points $A(2,3)$, $B(4,y)$ and $C(x,7)$ are such that $B$ is the midpoint of $\lbrack AC\rbrack$. Find $x$ and $y$.

33. Calculate the dot product $\overrightarrow{u} \cdot \overrightarrow{v}$, knowing that $||\overrightarrow{u}|| = 4$, $||\overrightarrow{v}|| = 3$ and the angle between them is $60^{\circ}$.

34. Find the equations of the lines parallel to $d:4x - 3y + 1 = 0$ at distance $2$ from it.

35. In triangle $ABC$, the point $A(1,1)$ and the equations of the perpendicular bisectors $x - y + 1 = 0$ and $2x + y - 4 = 0$ are known. Find the circumcenter of the triangle.

Level 3 – Deepening and consolidation (integrated mathematical reasoning)

36. Consider the lines $d_{m}:(2m + 1)x + (m - 1)y + 3m - 2 = 0$, $m \in \mathbb{R}$. Show that all the lines $d_{m}$ pass through a fixed point $P$ and find its coordinates.

37. In $\bigtriangleup ABC$, the vertices $A(2,4)$, $B( - 1,1)$ and $C(5, - 1)$ are given. Find the coordinates of the orthocenter $H$ of the triangle.

38. Find the coordinates of the reflection of the point $A(1,5)$ across the line $d:2x - y + 1 = 0$.

39. Let $A(1,2)$ and $B(3,0)$. Find the coordinates of the point $C$ on the first angle bisector ($y = x$) such that the area of triangle $ABC$ equals $4$.

40. Let the points $A(2,0)$ and $B(0,4)$. Find the coordinates of the points $M$ on the $Ox$ axis for which the angle $\widehat{AMB}$ is a right angle ($90^{\circ}$).

41. Find $a \in \mathbb{R}$ such that the vectors $\overrightarrow{u} = (a + 1)\overrightarrow{i} + 2\overrightarrow{j}$ and $\overrightarrow{v} = 3\overrightarrow{i} + (a - 1)\overrightarrow{j}$ form an obtuse angle.

42. In triangle $ABC$, the point $M$ divides side $BC$ in the ratio $\frac{BM}{MC} = 2$. Express the vector $\overrightarrow{AM}$ in terms of the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

43. Let $ABCD$ be any quadrilateral. Show using vectors that $\overrightarrow{AB} + \overrightarrow{CD} = \overrightarrow{AD} + \overrightarrow{CB}$.

44. Find the coordinates of the orthogonal projection of the origin $O(0,0)$ onto the line $d:3x - 4y + 25 = 0$.

45. Consider the lines $d_{1}:y = 2x + 1$, $d_{2}:y = - x + 4$ and $d_{3}:y = mx + 3$. Find $m \in \mathbb{R}$ such that the three lines are concurrent.

46. Consider the points $A(a,0)$, $B(0,b)$ with $a,b > 0$ such that the area of triangle $OAB$ is $6$. Knowing that the line $AB$ has slope $m = - \frac{3}{2}$, find $a$ and $b$.

47. Show that a point $P(x,y)$ equally distant from the lines $d_{1}:3x - 4y + 1 = 0$ and $d_{2}:5x + 12y - 2 = 0$ lies on one of the angle bisectors of the angles formed by the two lines, and write the equations of these bisectors.

48. Consider the position vectors of the vertices of $\bigtriangleup ABC$: $\overrightarrow{r_{A}} = \overrightarrow{i} + 2\overrightarrow{j}$, $\overrightarrow{r_{B}} = 4\overrightarrow{i} + 6\overrightarrow{j}$, $\overrightarrow{r_{C}} = 7\overrightarrow{i} - 1\overrightarrow{j}$. Find the magnitude of the vector $\overrightarrow{v} = \overrightarrow{r_{A}} + \overrightarrow{r_{B}} - 2\overrightarrow{r_{C}}$.

49. Find the equation of the line passing through the intersection point of the lines $x - y + 1 = 0$ and $2x + y - 4 = 0$ and parallel to the first angle bisector.

50. In $\bigtriangleup ABC$, the point $A(1,3)$ and the equations of the perpendicular bisectors of the sides $AB$ and $AC$: $x + y - 3 = 0$ and $x - 2y + 1 = 0$, respectively, are given. Find the coordinates of the vertices $B$ and $C$.