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 $\vec{A}$ and $c \vec{A}$ .

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 $B C = \vec{A}$ and $D E = c \vec{A}$

Use vector laws of addition

Now, using vector laws of addition

$\vec{A B} + \vec{B C} = \vec{A C} \vec{A D} + \vec{D E} = \vec{A E}$

Using the above two equations:

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

Prove that E is mid point of AC

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

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

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

Hence, $\vec{A E} = \frac{1}{2} \vec{A C}$ .

Therefore, E is the mid point of AC.

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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 $\vec{A}$ and $c \vec{A}$ .

Short Answer

Step by step solution

Given

Use vector laws of addition

Prove that E is mid point of AC

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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→and cA→.

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

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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 $\vec{A}$ and $c \vec{A}$ .