How many practice questions are available for Mid-Point Theorem?

SparkEd Math offers 60+ practice questions for Mid-Point Theorem across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Mid-Point Theorem content free on SparkEd Math?

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How does SparkEd Math teach Mid-Point Theorem differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Mid-Point Theorem problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Mid-Point Theorem content aligned with the ICSE syllabus?

Yes, all Mid-Point Theorem questions and lessons on SparkEd Math are fully aligned with the Indian Certificate of Secondary Education (ICSE) syllabus for Class 9. The content follows the official curriculum structure and exam pattern.

Mid-Point Theorem — Class 9 ICSE (2025-26)

About Mid-Point Theorem — Class 9 ICSE

Prove and apply the mid-point theorem and its converse in triangles and quadrilaterals. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 9). On this page you can practice 56 questions across three difficulty levels — 20 easy, 18 medium, and 18 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Mid-Point Theorem

Introduction to Mid-Point Theorem: The Basics
The Mid-Point Theorem: Statement and Proof
The Converse of Mid-Point Theorem: Understanding its Logic
Advanced Applications: Triangles and Quadrilaterals
Summary, Connections, and Exam Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Mid-Point Theorem — solved examples for Class 9 ICSE

Example 1easy

Which of the following statements correctly describes the Mid-Point Theorem for a triangle ABC, where D and E are mid-points of AB and AC respectively?

A)DE is parallel to BC and DE = BC
B)DE is perpendicular to BC and DE = (1/2) × BC
C)DE is parallel to AC and DE = (1/2) × BC
D)DE is parallel to BC and DE = (1/2) × BC

Step-by-step solution

The Mid-Point Theorem states that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of the length of the third side.
In triangle ABC, if D and E are mid-points of AB and AC respectively, then the segment DE must be parallel to BC and its length must be half of BC.
$DE || BC and DE = (1/2) × BC$

Answer: DE is parallel to BC and DE = (1/2) × BC

Example 2medium

In ΔABC, D and E are the mid-points of sides AB and AC respectively. If the perimeter of ΔADE is 15 cm and DE = 6 cm, find the perimeter of ΔABC.

A)30 cm
B)24 cm
C)27 cm
D)33 cm

Step-by-step solution

By the Mid-Point Theorem, DE is parallel to BC and DE = 1/2 BC. Given DE = 6 cm, so BC = 2 × 6 = 12 cm.
Since D and E are mid-points, AD = 1/2 AB and AE = 1/2 AC.
The perimeter of ΔADE = AD + DE + AE = 15 cm. Substituting AD = 1/2 AB, AE = 1/2 AC and DE = 6 cm: (1/2 AB) + 6 + (1/2 AC) = 15.
1/2 (AB + AC) = 15 - 6 = 9 cm. So, AB + AC = 18 cm. The perimeter of ΔABC = AB + BC + AC = (AB + AC) + BC = 18 cm + 12 cm = 30 cm.

Answer: 30 cm

Example 3hard

In triangle ABC, D is the midpoint of side AB and E is the midpoint of side AC. A line segment is drawn from D parallel to BE, intersecting AC at F. If AC = 12 cm, what is the length of AF?

A)3 cm
B)4 cm
C)6 cm
D)8 cm

Step-by-step solution

In ΔABC, E is the midpoint of AC. So, AE = EC = AC/2 = 12/2 = 6 cm.
Consider ΔABE. D is the midpoint of AB (given). DF is drawn parallel to BE (given).
By the Converse of Mid-Point Theorem in ΔABE, if a line through the midpoint of one side (D on AB) is parallel to another side (BE), it must bisect the third side (AE). Thus, F is the midpoint of AE.
Since F is the midpoint of AE, AF = AE/2 = 6/2 = 3 cm.

Answer: 3 cm

Practice questions on Mid-Point Theorem

Q1.easy
In triangle PQR, M and N are the mid-points of sides PQ and PR respectively. If QR = 14 cm, what is the length of MN?
1. A)7 cm
2. B)14 cm
3. C)28 cm
4. D)3.5 cm
Show answer
Answer: 7 cm
Hint: The segment connecting the mid-points of two sides is half the length of the third side.
Q2.easy
In triangle XYZ, A and B are the mid-points of sides XY and XZ respectively. Which statement about the line segment AB is true?
1. A)AB is parallel to YZ
2. B)AB is parallel to XY
3. C)AB is parallel to XZ
4. D)AB is perpendicular to YZ
Show answer
Answer: AB is parallel to YZ
Hint: Remember the parallelism aspect of the Mid-Point Theorem. The segment connects midpoints of two sides, so it's parallel to the *third* side.
Q3.easy
For a line segment drawn from the mid-point D of side AB of triangle ABC to a point E on AC to bisect side AC at E, which additional condition is *necessary* to apply the Converse of Mid-Point Theorem?
1. A)∠ADE = ∠B
2. B)∠ABC = 90°
3. C)DE must be parallel to BC
4. D)AC must be equal to BC
Show answer
Answer: DE must be parallel to BC
Hint: The Converse of Mid-Point Theorem requires a line starting from a midpoint and being parallel to another side. The conclusion is that it bisects the third side.
Q4.medium
In ΔPQR, S and T are mid-points of PQ and PR respectively. If ∠PST = 70° and ∠QPR = 40°, find the measure of ∠PQR.
1. A)60°
2. B)70°
3. C)80°
4. D)90°
Show answer
Answer: 70°
Hint: The Mid-Point Theorem states that the line segment connecting the midpoints is parallel to the third side. Think about the angles formed by parallel lines.
Q5.medium
In ΔABC, D is the midpoint of AB. A line through D parallel to BC intersects AC at E. If AE = (3x - 1) cm and EC = (x + 3) cm, find the length of AC.
1. A)10 cm
2. B)8 cm
3. C)14 cm
4. D)12 cm
Show answer
Answer: 14 cm
Hint: Recall the Converse of the Mid-Point Theorem. What does it tell you about point E?
Q6.medium
ABCD is a quadrilateral. P, Q, R, S are mid-points of AB, BC, CD, DA respectively. If AC = 10 cm and BD = 12 cm, find the perimeter of PQRS.
1. A)20 cm
2. B)22 cm
3. C)24 cm
4. D)26 cm
Show answer
Answer: 22 cm
Hint: Apply the Mid-Point Theorem to triangles formed by the diagonals of the quadrilateral. Each side of PQRS will be half the length of a diagonal.
Q7.hard
In parallelogram ABCD, M is the midpoint of side AB. A line segment DM is extended to meet CB produced at point N. If DC = 10 cm, what is the length of BN?
1. A)5 cm
2. B)7.5 cm
3. C)15 cm
4. D)10 cm
Show answer
Answer: 10 cm
Hint: Look for congruent triangles. Consider triangles involving M and exploit properties of parallelograms and parallel lines.
Q8.hard
In a quadrilateral ABCD, P, Q, R, S are the midpoints of sides AB, BC, CD, DA respectively. If the area of quadrilateral PQRS is 36 cm², what is the area of quadrilateral ABCD?
1. A)72 cm²
2. B)54 cm²
3. C)108 cm²
4. D)144 cm²
Show answer
Answer: 72 cm²
Hint: Recall the relationship between the area of a quadrilateral and the area of the parallelogram formed by joining its midpoints.
Q9.hard
In triangle ABC, D is the midpoint of BC. E is the midpoint of AD. If BE is extended to meet AC at F, then what is the ratio AF : FC?
1. A)1 : 3
2. B)2 : 3
3. C)1 : 2
4. D)1 : 1
Show answer
Answer: 1 : 2
Hint: Draw a line through point D parallel to BF. This will allow you to apply the Converse of Mid-Point Theorem twice.

These are 9 of the 56 questions available for Mid-Point Theorem. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.