SLIDE 1
DAY 68 – DILATION AS A RESULT
OF MIDPOINT THEOREM
SLIDE 2 INTRODUCTION
We have seen that a line passing joining midpoints
- f two sides of a triangle is parallel to the third side
and it is half the length of the third side. A part of this idea of parallelism, where is yet another idea about transformations that come out from the same
- concept. It is all about dilation.
In this lesson, we are going to show how the drawing of this line leads to a line and a triangle which is a dilation of the original line and a triangle.
SLIDE 3
VOCABULARY
Dilation
A transformation where the object is increased or reduced in size while maintaining its shape.
SLIDE 4
The midpoint theorem states that a line joining midpoints of two sides of a triangle are parallel to the third side and that it is half the length of the third side. Consider ∆𝑇𝑈𝑉 below. Points A and B are midpoints of 𝑇𝑈 and 𝑇𝑉 respectively.
𝑉 𝑈 𝑇 𝐶 𝐵
SLIDE 5
In the figure below we have joined the midpoints of 𝑇𝑉 and 𝑇𝑈 with a straight line. From the statement of the midpoint theorem, line AB is parallel to side 𝑈𝑉 and its length is half the length of 𝑈𝑉.
𝑉 𝑈 𝑇 𝐶 𝐵
SLIDE 6 Drawing of line AB has led to formation of ∆𝑇𝐵𝐶. The length of side SB is half the length of side SU, and the length of side SA is half the length of side
- ST. We dilate ∆𝑇𝑈𝑉 with a scale factor of 1
2 about
point S, by multiplying the lengths of ST and SU by
1 2 to get triangle SBA.
𝑉 𝑈 𝑇 𝐶 𝐵
SLIDE 7 Multiplying this lengths by 1
2 leads to lengths
which stretches from point S to their midpoints. Thus, SU, ST and AB are dilated by a factor of 0.5 to SB, SA and TU respectively. Likewise, we may say SB, SA and TU are dilated by a factor of 2 to SU, ST and AB respectively.
𝑉 𝑈 𝑇 𝐶 𝐵
SLIDE 8 The triangle formed by the dilation is the same as the triangle that is formed by drawing a line joining the midpoints of sides ST and SU. In general, drawing a line joining midpoints of two sides of a triangle results in a dilation of that triangle with a scale factor of 1
2 about the vertex
included by those sides.
SLIDE 9 Example In the figure below, ∆𝐵𝐶𝐷 is dilated with a scale factor of 1
2 about point C to form ∆𝐵′𝐶′𝐷′. Show that
𝐵′𝐶′ is parallel to AB.
𝐵′ 𝐵 B 𝐶′ 𝐷
SLIDE 10 Solution Since the scale factor is 1
2,
𝐷𝐵′ =
1 2 𝐵𝐷
𝐷𝐶′ =
1 2 𝐶𝐷
Therefore points 𝐵′ and 𝐶′ are midpoints of sides AC and BC respectively. Since, the line joining the midpoints of two sides a triangle is parallel to the third side, line 𝐵′𝐶′ is parallel to line AB.
SLIDE 11
HOMEWORK In ∆𝑁𝑂𝑃 below, points A and B are midpoints of sides ON and OM respectively. If the length of MN is 2.8𝑗𝑜, what is the length of 𝐵𝐶?
𝑁 𝑂 𝑃 𝐵 𝐶
SLIDE 12
ANSWERS TO HOMEWORK
1.4𝑗𝑜
SLIDE 13
THE END
