Frank Solutions for Class 9 Maths Chapter 15 Mid - point and ... - Free Printable

The image you've uploaded contains two geometry problems based on the Mid-point Theorem and Intercept Theorems, along with their solutions. Let's go through both problems step by step, explain the reasoning, and verify the solutions.

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🔷 Problem 1:

> In ΔABC, D is the mid-point of AB and E is the mid-point of BC.
> Calculate:
> (i) DE, if AC = 8.6 cm
> (ii) ∠DEB, if ∠ACB = 72°

#### ✔ Solution Explanation:

We are given:
- D is the midpoint of AB
- E is the midpoint of BC
- So, DE connects the midpoints of sides AB and BC.

This is a classic case for the Mid-point Theorem:

> Mid-point Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half its length.

So, in ΔABC:
- Since D and E are midpoints of AB and BC respectively,
- Then, DE ∥ AC and DE = (1/2) × AC

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(i) Find DE, if AC = 8.6 cm

Using the Mid-point Theorem:
$$
DE = \frac{1}{2} \times AC = \frac{1}{2} \times 8.6 = 4.3\ \text{cm}
$$

✔ Answer: DE = 4.3 cm

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(ii) Find ∠DEB, if ∠ACB = 72°

We know:
- DE ∥ AC (from Mid-point Theorem)
- Line BC is a transversal cutting these parallel lines.

So, ∠DEB and ∠ACB are corresponding angles.

> When two parallel lines are cut by a transversal, corresponding angles are equal.

Thus:
$$
∠DEB = ∠ACB = 72^\circ
$$

✔ Answer: ∠DEB = 72°

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🔷 Problem 2:

> In ΔABC, AB = 12 cm and AC = 9 cm. M is the mid-point of AB. A straight line through M parallel to AC cuts BC at N. What is the length of MN?

#### ✔ Solution Explanation:

Given:
- AB = 12 cm → M is midpoint → AM = MB = 6 cm
- AC = 9 cm
- MN ∥ AC
- MN passes through M (midpoint of AB)

We need to find MN.

Let’s apply the converse of the Mid-point Theorem or use similar triangles.

But here's the key idea:

> If a line is drawn through the midpoint of one side of a triangle and is parallel to another side, then it bisects the third side.

So:
- M is midpoint of AB
- MN ∥ AC
→ Then, N must be the midpoint of BC (by the converse of the Mid-point Theorem)

Now, since MN joins the midpoints of AB and BC, and MN ∥ AC, we can again use the Mid-point Theorem:

> The segment joining midpoints of two sides is half the third side and parallel to it.

So:
$$
MN = \frac{1}{2} \times AC = \frac{1}{2} \times 9 = 4.5\ \text{cm}
$$

✔ Answer: MN = 4.5 cm

(Note: The solution in the image says “(9/2) cm” which is correct — that is 4.5 cm.)

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✔ Final Answers:

#### Problem 1:
(i) $ DE = 4.3\ \text{cm} $
(ii) $ ∠DEB = 72^\circ $

#### Problem 2:
$ MN = 4.5\ \text{cm} $

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📌 Key Concepts Used:

1. Mid-point Theorem:
- The line segment joining the midpoints of two sides of a triangle is:
- Parallel to the third side
- Half its length

2. Converse of Mid-point Theorem:
- If a line through the midpoint of one side is parallel to another side, it bisects the third side.

3. Corresponding Angles:
- When two lines are parallel, corresponding angles formed by a transversal are equal.

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