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Q20E

Expert-verifiedFound in: Page 470

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**Prove that the midpoint of the line segment between two constructible points is a constructible point.**** [Hint: Adapt the hint to Exercise ${\mathbf{17}}$** **.]**

The midpoint of the segment between two constructible points is a constructible point.

**Drawing precise lines, line segments, shapes, circles, and other figures with a ruler, compass, or protractor is known as geometric construction.**

- A point $(r,s)$ is constructible iff $r$ and $s$ are constructible.
- If $a,b,c$ with $c\ne 0$ are constructible then $a+b,a-b$ and $a/c$ are constructible.

The proving part:

Let $\mathbb{Q}\left[x\right]$ , $\text{\hspace{0.17em}}(\text{\hspace{0.17em}}{a}_{1},{a}_{2})$and $\text{\hspace{0.17em}}(\text{\hspace{0.17em}}{b}_{1},{b}_{2})$ are constructible points.

Then from $1$ **, **each $\text{\hspace{0.17em}}{a}_{1},{a}_{2},{b}_{1},{b}_{2}$ are constructible.

Form point$2$ in step 1^{st}, we infer that $\frac{{a}_{1}+{b}_{1}}{2}$ and $\frac{{a}_{2}+{b}_{2}}{2}$ are constructible numbers.

Then by reverse of point $1$ in step 1^{st}, the points $(\frac{{a}_{1}+{b}_{1}}{2},\text{\hspace{0.17em}}\frac{{a}_{2}+{b}_{2}}{2})$ is constructible.

Hence, the midpoint of the line segment between two constructible points is a constructible point.

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