Chapter 3 · Model TestAttempt first – solutions on the next page

Model Test Paper മാതൃകാ ചോദ്യപേപ്പർ

Set a timer for 1½ hours, keep the textbook closed, and write your answers in a notebook. Keep a ruler, compass and set squares ready for the construction questions. Then check your work on the solved paper, which also shows how marks are given.

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Kerala State Syllabus · Standard IX · Mathematics Part 1

MATHEMATICS – Chapter 3: Parallel Lines (സമാന്തരവരകൾ)

Time: 1½ hoursMaximum marks: 40
Instructions: Read each question carefully. Show your working – marks are given for correct steps even when the final answer is wrong. For constructions, describe the steps and draw a rough sketch; for proofs, name the theorem you use at each step. Section A: 8 × 1 = 8, Section B: 5 × 2 = 10, Section C: 4 × 3 = 12, Section D: 2 × 5 = 10.
SECTION A – Answer all questions8 × 1 = 8 marks
1.
Equally spaced parallel lines cut any line into parts.
(1)
2.
Three parallel lines cut one line into pieces of 3 cm and 5 cm. They cut a second line into pieces of 6 cm and cm.
(1)
3.
In triangle ABC, a line parallel to BC meets AB at P and AC at Q. If AP : PB = 2 : 3 and AC = 10 cm, then AQ = cm.
(1)
4.
The line joining the midpoints of two sides of a triangle is to the third side and of it.
(1)
5.
The point where the three medians of a triangle meet is called the
(a) orthocentre(b) centroid(c) circumcentre(d) incentre
(1)
6.
The centroid divides each median, measured from the vertex, in the ratio
(a) 1 : 1(b) 2 : 1(c) 3 : 1(d) 1 : 2
(1)
7.
The circumcentre of a right triangle is
(a) the right-angle vertex(b) the midpoint of the hypotenuse(c) the centroid(d) outside the triangle
(1)
8.
The triangle formed by joining the midpoints of the sides of a triangle has perimeter 9 cm. The perimeter of the original triangle is cm.
(1)
SECTION B – Answer all questions5 × 2 = 10 marks
9.
Describe, with a rough sketch, how to divide a 7 cm line into 3 equal parts using parallel lines. What is the length of each part?
(2)
10.
A 12 cm line is to be divided in the ratio 1 : 3. Describe the construction and write the lengths of the two parts.
(2)
11.
In triangle ABC, AB = 8 cm and AC = 6 cm. A line parallel to BC meets AB at P and AC at Q. If AP = 2 cm, find AQ and QC.
(2)
12.
The diagonals of a quadrilateral are 10 cm and 6 cm long. The midpoints of its sides are joined in order. What are the lengths of the sides of the quadrilateral so formed, and what is its perimeter?
(2)
13.
  1. In triangle ABC the median AD is 12 cm long and G is the centroid. Find AG and GD.
  2. The hypotenuse of a right triangle is 10 cm. How far is the circumcentre from each vertex?
(2)
SECTION C – Answer all questions4 × 3 = 12 marks
14.
Prove that in any triangle, the line drawn through the midpoint of one side parallel to a second side bisects the third side.
(3)
15.
In the trapezium ABCD, AB ∥ DC and the diagonals AC and BD meet at O.
  1. Prove that AO/OC = BO/OD.
  2. If AO = 6 cm, OC = 4 cm and BD = 15 cm, find BO and OD.
(3)
16.
A triangle has perimeter 12 cm and its sides are in the ratio 3 : 4 : 5.
  1. Explain how to divide a 12 cm line in the ratio 3 : 4 : 5 using parallel lines, and why the construction gives the correct ratio.
  2. Write the lengths of the three sides, draw a rough sketch of the triangle, and say what kind of triangle it is.
(3)
17.
In a right triangle the perpendicular sides are 6 cm and 8 cm. The right-angle vertex and the midpoints of the three sides are joined to form a quadrilateral, as in the figure.
  1. Prove that this quadrilateral is a rectangle.
  2. Find its area and compare it with the area of the triangle.
(3)
SECTION D – Answer all questions2 × 5 = 10 marks
18.
Prove that the three medians of a triangle pass through one point, and that this point divides each median in the ratio 2 : 1 from the vertex.
(5)
19.
  1. Prove that the quadrilateral formed by joining the midpoints of the sides of any quadrilateral, in order, is a parallelogram.
  2. If the diagonals of the original quadrilateral are equal and perpendicular to each other, what special kind of parallelogram is formed? Give reasons.
  3. The diagonals of a quadrilateral are perpendicular and each is 8 cm long. Find the area of the quadrilateral formed by joining the midpoints of its sides.
(5)

— End of the paper —

📝 After the test

Check your answers on the next page. Give yourself marks honestly using the marking hints there. Anything below 28/40 means the chapter needs one more round – the solved paper tells you exactly which page to revisit for each question.

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