Theorems Related with Area

Maths, SSC-KPK Class 9. This chapter is split into 30 topics and has 663 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Theorem 16.2 in numerical problems (Part 1) 21 questions
2. Ratios of areas for equal bases or equal altitudes 30 questions
3. Theorem 16.4 in numerical problems (Part 1) 20 questions
4. Perpendicular distance between parallel lines 20 questions
5. Exercise 16.1 Q2-Q3: fractions of a parallelogram and proving parallelism 21 questions
6. Theorem 16.3: triangles on the same base 20 questions
7. Area of a triangle from base and altitude (Part 2) 24 questions
8. Altitude and base of a triangle 21 questions
9. Medians, midpoints and equal areas (Review Exercise 16 Q2-Q5) 30 questions
10. Theorem 16.3 in numerical problems (Part 1) 24 questions
11. Proof of Theorem 16.2 20 questions
12. Exercise 16.1 Q1: equal-area triangles in △ABC with PQ ∥ BC 20 questions
13. Area of a parallelogram: base, altitude and unknown dimensions (Part 1) 23 questions
14. Theorem 16.1: statement and figure 20 questions
15. Theorem 16.4 in numerical problems (Part 2) 20 questions
16. Area of a parallelogram: base, altitude and unknown dimensions (Part 2) 22 questions
17. Theorem 16.2 in numerical problems (Part 2) 20 questions
18. Area of a triangle from base and altitude (Part 1) 24 questions
19. Equal area versus congruence 20 questions
20. Theorem 16.1 in numerical problems (Part 1) 22 questions
21. Theorem 16.2: statement and figure 20 questions
22. Triangles and parallelograms: diagonals and halves (Part 1) 21 questions
23. Review Exercise 16 multiple-choice questions 23 questions
24. Areas of rectangles, squares and quadrilaterals using triangles 30 questions
25. Proof of Theorem 16.3 20 questions
26. Proof of Theorem 16.1 20 questions
27. Theorem 16.1 in numerical problems (Part 2) 22 questions
28. Theorem 16.4: triangles on equal bases 20 questions
29. Triangles and parallelograms: diagonals and halves (Part 2) 21 questions
30. Theorem 16.3 in numerical problems (Part 2) 24 questions

Sample questions: Theorems Related with Area

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. If two figures lie between the same parallel lines they are said to have the same

• A. angle
• B. base
• C. area
• D. altitude(correct answer)

Answer: D.Lying between the same parallels means equal altitudes (the book writes '(or of the same altitude)').

Q2. In the proof of 16.1 subtracting the two congruent triangles from the whole figure leaves

• A. two congruent triangles
• B. two trapezia
• C. the parallelograms ABGH and ABCD respectively(correct answer)
• D. a rectangle

Answer: C.Whole figure minus the left triangle is one parallelogram and minus the right triangle is the other.

Q3. A triangle and a parallelogram on the same base between the same parallels have areas related as

• A. triangle = half parallelogram(correct answer)
• B. triangle = one third parallelogram
• C. triangle = parallelogram
• D. triangle = twice parallelogram

Answer: A.The diagonal splits the parallelogram into two equal triangles each equal in area to the given triangle.

Q4. Why are ABCD and ABGH equal in area in the proof of 16.3?

• A. They are on the same base and between the same parallels(correct answer)
• B. They are rectangles
• C. They have equal angles
• D. They are congruent

Answer: A.Theorem 16.1. The correct answer is: They are on the same base and between the same parallels.

Q5. In Exercise 16.1 Q1, why are △PBC and △QBC equal in area?

From your book: Exercise 16.1 (page 300) Full page Open book

• A. They are congruent
• B. They are similar
• C. Same base BC and between the same parallel lines PQ and BC(correct answer)
• D. They are right-angled

Answer: C.Theorem 16.3. The correct answer is: Same base BC and between the same parallel lines PQ and BC.

Q6. PQRS is a parallelogram. A and B are the midpoints of PQ and PS. The area of △PAB is what fraction of the parallelogram?

• A. 1/4
• B. 3/8
• C. 1/2
• D. 1/8(correct answer)

Answer: D.△PQS is half the parallelogram and △PAB is a quarter of △PQS (sides halved), so 1/8 (Exercise 16.1 Q2 i).

Q7. PQRS is a parallelogram with A and B the midpoints of PQ and PS. The area of △QBR is what fraction of the parallelogram?

• A. 1/4
• B. 1/2(correct answer)
• C. 1/8
• D. 3/8

Answer: B.△QBR has base QR and lies between the parallel lines PS and QR, so it is half the parallelogram (Exercise 16.1 Q2 ii).

Q8. In parallelogram PQRS, A and B are midpoints of PQ and PS. The area of △AQR is what fraction of PQRS?

• A. 1/4(correct answer)
• B. 3/8
• C. 1/8
• D. 1/2

Answer: A.△AQR has base AQ = half of PQ and altitude equal to that of the parallelogram: ½ × (1/2) = 1/4.

Q9. Exercise 16.1 Q3: ABCD is a quadrilateral whose diagonals AC and BD meet at E. If △ABE and △CDE are equal in area then

• A. AD ∥ BC(correct answer)
• B. AD = BC
• C. AB ∥ CD
• D. AC = BD

Answer: A.Adding △ADE gives △ABD = △ACD on the same base AD, so B and C are at equal distances from AD: AD ∥ BC.

Q10. D and E are the midpoints of AB and AC of △ABC. The areas of △ADE, △BDE and △CDE are

• A. in ratio 1 : 2 : 3
• B. in ratio 1 : 1 : 2
• C. all equal(correct answer)
• D. unequal

Answer: C.DE is a median of △ABE (D midpoint of AB) so △ADE = △BDE, and likewise △ADE = △CDE (E midpoint of AC) (Review Exercise 16 Q3).