Enhanced Curriculum Support

SecA

1. Find the area of the triangle ABC with A (1, - 4) and mid-points of sides through A being (2, - 1) and (0, -1).

2. Find the ratio in which the point P (3/4, 5/12) divides the line segment joining the points A (1/2, 3/2) and B (2, -5).

3. If A(-4, 8), B(-3, -4), C(0, -5) and D(5, 6) are the vertices of a quadrilateral ABCD, find its area.

4. Find the coordinates of the point P dividing the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2:1.

5. If the distance between the points (4, 𝑘) and (1,0) is 5, then what can be the possible values of 𝑘?

6. Find the area of a triangle whose vertices are (3, 0), (7, 0).and (8, 4).

7. 𝐴𝐵𝐶𝐷 is a rectangle whose three vertices are 𝐵(4, 0), 𝐶(4, 3) and 𝐷(0, 3). The length of one of its diagonals is

8. If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (−2, 5), then the coordinates of the other end of the diameter are :

9. The area of a triangle whose vertices are (5, 0), (8, 0) and (8, 4) (in sq. units) is

10. The point 𝑃 which divides the line segment joining the points 𝐴(2, −5) and 𝐵(5,2) in the ratio 2 ∶ 3 lies in the quadrant

SecB

1. If two vertices of an equilateral triangle are (3, 0) and (6, 0), find the third vertex.

2. . If two adjacent vertices of a parallelogram are (3,2) and (−1,0) and the diagonals intersect at (2, −5), then find the coordinates of the other two vertices.

3. Find the value of 𝑝 for which the points (−1,3), (2, 𝑝) and (5, −1) are collinear.

4. Find the ratio in which the point (−3, 𝑘) divides the line-segment joining the points (−5, −4) and (−2, 3). Also find the value of 𝑘.

5. Prove that the points (2, −2), (−2, 1) and (5, 2) are the vertices of a right angled triangle. Also find the area of this triangle.

6. Prove that the points (3, 0), (6, 4) and (−1, 3) are the vertices of a right angled isosceles triangle.

7. Find the ratio in which the line segment joining the points (1, −3) and (4, 5) is divided by 𝑥-axis.

8. Find the ratio in which the y-axis divides the line segment joining the points (-4,-6) and (10, 12). Also find the coordinates of the point of division.

9. If a point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), then find the value of p.

10. If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (-2, 5), then what are the coordinates of the other end of the diameter?

SecC

1. The vertices of a triangle are 𝐴(−1,3), 𝐵(1, −1) and 𝐶(5,1). Find the length of the median through the vertex 𝐶.

2. Find the coordinates of the points of trisection of the line segment joining the points (3, −2) and (−3, −4).

3. Show that ∆𝐴𝐵𝐶 with vertices 𝐴(−2,0), 𝐵(0,2) and 𝐶(2,0) is similar to ∆ 𝐷𝐸𝐹 with vertices 𝐷(−4,0), 𝐹(4,0) and 𝐸(0,4)

4. Point 𝐴 lies on the line segment 𝑃𝑄 joining 𝑃(6, −6) and 𝑄(−4, −1) in such a way that 𝑃𝐴/𝑃𝑄=2/5. If point 𝑃 also lies on the line 3𝑥 + 𝑘(𝑦 + 1) = 0, find the value of 𝑘.

5. Prove that the diagonals of a rectangle 𝐴𝐵𝐶𝐷, with vertices 𝐴(2, −1), 𝐵(5, −1), 𝐶(5, 6) and 𝐷(2, 6), are equal and bisect each other.

6. Find a point 𝑃 on the 𝑦-axis which is equidistant from the points 𝐴(4, 8) and 𝐵(−6, 6). Also find the distance. 𝐴𝑃.

7. If the vertices of a triangle are (1, −3), (4, 𝑝) and (−9, 7) and its area is 15 𝑠𝑞. Units, find the value(s) of p.

8. The line segment joining the points 𝐴(2, 1) and 𝐵(5, −8) is trisected at the point 𝑃 and 𝑄 such that 𝑃 is nearer to 𝐴. If 𝑃 also lies on the line given by 2𝑥 − 𝑦 + 𝑘 = 0, find the value of 𝑘.

Show that 𝐴(−3, 2), 𝐵(−5, −5), 𝐶(2, −3) and 𝐷(4, 4)are the vertices of a rhombus

The mid-points of the sides of triangle are (3, 4), (4, 6) and (5, 7). Find the coordinates of the vertices of the triangle.

If 𝑃 divides the join of 𝐴(−2, −2) and 𝐵(2, −4) such that 𝐴𝑃/𝐴𝐵 =3/7, find the coordinates of 𝑃.

SecD

1. If 𝑃(9𝑎 − 2, −𝑏) divides the line segment joining 𝐴(3𝑎 + 1, −3) and 𝐵(8𝑎, 5) in the ratio 3 ∶ 1. Find the values of 𝑎 & 𝑏.

2. Find the coordinates of the points which divide the line segment joining 𝐴(2, −3) and 𝐵(−4, −6) into three equal parts.

3. Prove that the area of a triangle with vertices (𝑡, 𝑡 − 2), (𝑡 + 2, 𝑡 + 2) and (𝑡 + 3, 𝑡) is independent of t

4. The base 𝐵𝐶 of an equilateral triangle 𝐴𝐵𝐶 lies on 𝑦–axis. The coordinates of point 𝐶 are (0, −3). The origin is the mid–point of the base. Find the coordinates of the points 𝐴 and 𝐵. Also find the coordinates of another point 𝐷 such that 𝐵𝐴𝐶𝐷 is a rhombus.

5. The points 𝐴(1, −2), 𝐵(2,3), 𝐶(𝑘, −2) and 𝐷(−4, −3) are the vertices of a parallelogram. Find the value of 𝑘 and the altitude of the parallelogram corresponding to the base 𝐴𝐵.

6. If 𝐴(4, 2), 𝐵(7, 6) and 𝐶(1, 4) are the vertices of a ∆ 𝐴𝐵𝐶 and 𝐴𝐷 is its median, prove that the median 𝐴𝐷 divides ∆ 𝐴𝐵𝐶 into two triangles of equal areas.

7. Find the values of 𝑘 so that the area of the triangle with vertices (1, −1), (−4, 2𝑘) and (−𝑘, −5) is 24 sq. Units

8. If 𝐴(4, 2), 𝐵(7, 6) and 𝐶(1, 4) are the vertices of a ∆ 𝐴𝐵𝐶 and 𝐴𝐷 is its median, prove that the median 𝐴𝐷 divides ∆ 𝐴𝐵𝐶 into two triangles of equal areas.

9. If 𝐴(−4, 8), 𝐵(−3, −4), 𝐶(0, −5) and 𝐷(5, 6) are the vertices of a quadrilateral 𝐴𝐵𝐶𝐷, find its area

Problem1

Situation: The city council plans to build a library that is accessible to three neighborhoods. The neighborhoods are represented by points A(2,3), B(6,7), and C(4,5) on a coordinate plane. The council wants to place the library at a point equidistant from all three neighborhoods to ensure fairness and accessibility.

1. Determine the coordinates where the library should be placed.

2. How does placing the library equidistant from all neighborhoods reflect the value of fairness and inclusivity in urban planning?

Problem2

Situation: A triangular park is being developed in a town to increase green spaces. The vertices of the triangle representing the park are at points A(1,2), B(5,8), and C(9,2). The park authorities decide to plant trees along the median from A to BC to symbolize the balance between development and nature.

1. Find the equation of the median from A to BC

2. How does this decision reflect the value of environmental conservation?

Problem3

Situation: A sports club wants to set up a running track that is equidistant from three key points A(3,4), B(7,10), and C(11,4) on a coordinate plane, where sports facilities like a gym, a swimming pool, and a basketball court are located. The running track should be accessible to all members equally.

1. Find the coordinates of the center of the running track?

2. How does this setup promote equality among the members?

Problem4

Situation: A school is planning to place a new resource center that is equidistant from three existing blocks (A, B, C) in the school compound. The blocks are located at A(-2,1), B(4,5), and C(6,-3) on the coordinate plane. The resource center should be accessible to all students equally, regardless of their block.

Determine the coordinates of the point where the resource center should be located. Discuss how this decision promotes educational equity.

Q1

1. ∆ ABC with vertices A (–2, 0), B (2, 0) and C (0, 2) is similar to ∆ DEF with vertices D (–4, 0) E (4, 0) and F (0, 4).

2. Point P (– 4, 2) lies on the line segment joining the points A (– 4, 6) and B (– 4, – 6).

Q2

1. Point P (5, –3) is one of the two points of trisection of the line segment joining the points A (7, – 2) and B (1, – 5).

2. The points A (–1, –2), B (4, 3), C (2, 5) and D (–3, 0) in that order form a rectangle

3. Name the type of triangle formed by the points A (–5, 6), B (–4, –2) and C (7, 5)

Q3

1.The point A (2, 7) lies on the perpendicular bisector of line segment joining the points P (6, 5) and Q (0, – 4).

2. Point P (0, 2) is the point of intersection of y–axis and perpendicular bisector of line segment joining the points A (–1, 1) and B (3, 3).

3. If P (9a – 2, –b) divides line segment joining A (3a + 1, –3) and B (8a, 5) in the ratio 3 : 1, find the values of a and b.

Q4

1. Find the coordinates of the point Q on the x–axis which lies on the perpendicular bisector of the line segment joining the points A (–5, –2) and B(4, –2). Name the type of triangle formed by the points Q, A and B.

2. Find the value of a, if the distance between the points A (–3, –14) and B (a, –5) is 9 units.

3. The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, –9) and has diameter 10 sqrt2 units.

Q5

1. The line segment joining the points A (3, 2) and B (5,1) is divided at the point P in the ratio 1:2 and it lies on the line 3x – 18y + k = 0. Find the value of k.

2.The points A (2, 9), B (a, 5) and C (5, 5) are the vertices of a triangle ABC right angled at B. Find the values of a and hence the area of ∆ABC.

3. The mid-points D, E, F of the sides of a triangle ABC are (3, 4), (8, 9) and (6, 7). Find the coordinates of the vertices of the triangle.

Questions

1. The distance of the point P (2, 3) from the x-axis is?

2. The area of a triangle with vertices A (3, 0), B (7, 0) and C (8, 4) is?

3. The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is

4. The distance of the point P (–6, 8) from the origin is

5. AOBC is a rectangle whose three vertices are vertices A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is

6. The points (–4, 0), (4, 0), (0, 3) are the vertices of a

7. The point which divides the line segment joining the points (7, –6) and (3, 4) in ratio 1 : 2 internally lies in the

8.The point which lies on the perpendicular bisector of the line segment joining the points A (–2, –5) and B (2, 5) is?

9. The fourth vertex D of a parallelogram ABCD whose three vertices are A (–2, 3), B (6, 7) and C (8, 3) is

10. If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then

11. The perpendicular bisector of the line segment joining the points A (1, 5) and B (4, 6) cuts the y-axis at

12. The points (4, 5), (7, 6) and (6, 3) are collinear

13. Point P (0, –7) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A (–1, 0) and B (7, –6).

14. If the points A (1, 2), O (0, 0) and C (a, b) are collinear, then

15. The area of a triangle with vertices (a, b + c), (b, c + a) and (c, a + b) is

16. The points A (–1, 0), B (3, 1), C (2, 2) and D (–2, 1) are the vertices of a parallelogram

Question 1

Mr. Mohanlal purchased a land which is in the shape of a quadrilateral. Its four corner points are given in the graph.

Based on your understanding of the above case study, answer all the five questions below:

1. The distance between point A and point D is

2. The co-ordinate of intersecting point of the diagonals is

3. The quadrilateral formed after joining A, B, C and D is

__{.m-red}4. If (1,2), (4, y), (x,6) and (3,5) are the vertices of the given parallelogram taken in order the value of x and y respectively are and

5. The distance of point A from the origin is

Question 2

Sarita has a kitchen garden of size 10m x 10m in her bungalow. She wants to grow vegetables that are used daily in her kitchen. She has divided her whole kitchen garden into a 10 x 10 grid as shown in the figure. For that she has put manure in the soil to increase the output. She has planted a tomato plant at A, a coriander plant at C and a green chilli plant at B. She invited her friend Sita to show her the kitchen garden. Sita says that saplings at A, B and C form an equilateral triangle.

Based on your understanding of the above case study, answer all the five questions below:

1. The co-ordinates of the points B and C are

2. Name the type of triangle formed by joining the points of A, B and C

3. The point which is equidistant from the point A, B and C is

4. If the co-ordinates of a point D marked on AB is(2,223) then the ratio of AD and BD is

5. The c-ordinates of mid-point of BC is