Q7P A line through the midpoint of o... [FREE SOLUTION]

Chapter 3: Q7P (page 105)

A line through the midpoint of one side of a triangle and parallel to a second side bisects the third side. Hint: Call parallel vectors $\overset{鈫�}{A}$ and $c \overset{鈫�}{A}$ .

Short Answer

Expert verified

It has been proved that a line through the midpoint of one side of a triangle and parallel to a second side bisects the third side.

Step by step solution

Given

A triangle ABC, D is the midpoint of AB, E is a point on AC such that DE is parallel to BC.

Here, $A D = \frac{1}{2} A B$ .

Let $B C = \overset{鈫�}{A}$ and $D E = c \overset{鈫�}{A}$

Use vector laws of addition

Now, using vector laws of addition

$\overset{鈫�}{A B} + \overset{鈫�}{B C} = \overset{鈫�}{A C} \overset{鈫�}{A D} + \overset{鈫�}{D E} = \overset{鈫�}{A E}$

Using the above two equations:

$\overset{鈫�}{D E} = \overset{鈫�}{A E} - \overset{鈫�}{A D} c \overset{鈫�}{A} = \overset{鈫�}{A E} - \frac{1}{2} \overset{鈫�}{A B} c (\overset{鈫�}{A C} - \overset{鈫�}{A B}) = \overset{鈫�}{A E} - \frac{1}{2} \overset{鈫�}{A B} c \overset{鈫�}{A C} - \overset{鈫�}{A E} = (c - \frac{1}{2}) \overset{鈫�}{A B}$

Prove that E is mid point of AC

Now, E lies on $\overset{鈫�}{A C}$ so $\overset{鈫�}{A E}$ can be written as the product of real number and vector $\overset{鈫�}{A C}$ .

Thus, $c - \frac{1}{2}$ must be zero.

Thus $c = \frac{1}{2}$ .

Hence, $\overset{鈫�}{A E} = \frac{1}{2} \overset{鈫�}{A C}$ .

Therefore, E is the mid point of AC.

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91影视

A line through the midpoint of one side of a triangle and parallel to a second side bisects the third side. Hint: Call parallel vectors $\overset{鈫�}{A}$ and $c \overset{鈫�}{A}$ .

Short Answer

Step by step solution

Given

Use vector laws of addition

Prove that E is mid point of AC

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91影视

A line through the midpoint of one side of a triangle and parallel to a second side bisects the third side. Hint: Call parallel vectors A鈫�and cA鈫�.

Short Answer

Step by step solution

Given

Use vector laws of addition

Prove that E is mid point of AC

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Circular Motion and Gravitation

Energy Physics

Engineering Physics

Electromagnetism

Space Physics

Study anywhere. Anytime. Across all devices.

A line through the midpoint of one side of a triangle and parallel to a second side bisects the third side. Hint: Call parallel vectors $\overset{鈫�}{A}$ and $c \overset{鈫�}{A}$ .