Day 68 – Dilation as a result of midpoint theorem

Introduction We have seen that a line passing joining midpoints of 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.

Vocabulary Dilation A transformation where the object is increased or reduced in size while maintaining its shape.

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 ∆𝑆𝑇𝑈 below. Points A and B are midpoints of 𝑆𝑇 and 𝑆𝑈 respectively. 𝑈 𝑇 𝑆 𝐵 𝐴

In the figure below we have joined the midpoints of 𝑆𝑈 and 𝑆𝑇 with a straight line. From the statement of the midpoint theorem, line AB is parallel to side 𝑇𝑈 and its length is half the length of 𝑇𝑈. 𝑈 𝑇 𝑆 𝐵 𝐴

Drawing of line AB has led to formation of ∆𝑆𝐴𝐵 Drawing of line AB has led to formation of ∆𝑆𝐴𝐵. 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 ∆𝑆𝑇𝑈 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. 𝑈 𝑇 𝑆 𝐵 𝐴

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. 𝑈 𝑇 𝑆 𝐵 𝐴

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.

Example In the figure below, ∆𝐴𝐵𝐶 is dilated with a scale factor of 1 2 about point C to form ∆ 𝐴 ′ 𝐵 ′ 𝐶 ′ . Show that 𝐴′𝐵 ′ is parallel to AB. 𝐴′ 𝐴 B 𝐵′ 𝐶

Solution Since the scale factor is 1 2 , 𝐶𝐴′= 1 2 𝐴𝐶 𝐶 𝐵 ′ = 1 2 𝐵𝐶 Therefore points 𝐴′ and 𝐵′ 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 𝐴 ′ 𝐵 ′ is parallel to line AB.

homework In ∆𝑀𝑁𝑂 below, points A and B are midpoints of sides ON and OM respectively. If the length of MN is 2.8𝑖𝑛, what is the length of 𝐴𝐵? 𝑀 𝑁 𝑂 𝐴 𝐵

Answers to homework 1.4𝑖𝑛

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