Q28. - Geometry

Question

Draw a segment and let its length be $1$.

$\left(a\right).$ Construct a segment of length $\sqrt{5}$.

$\left(b\right).$ Construct a segment of length $\frac{1}{2}+\frac{\sqrt{5}}{2}$, or $\frac{1+\sqrt{5}}{2}$.

$\left(c\right).$ Construct a golden rectangle whose sides are in the ratio $1:\frac{1+\sqrt{5}}{2}$.

$\left(a\right)$

Step-by-Step Solution

Answer

The figure is,

Step 1. Given information.

Draw a segment with length $1$.

Step 2. Follow the ways.

To draw a segment of length $\sqrt{5}$, follow these simple ways-

Draw a segment with length of $1$.

Double it in the horizontal direction and take it as it is in vertical direction.

Step 3. Draw the figure.

Now complete the triangle, and the closing side will have length $\sqrt{5}$. 

since ($\because \sqrt{5}=\sqrt{2^2+1^2}$)

The figure is drawn below,

Therefore, the required figure drawn above.

$\left(b\right)$

The figure is,

Step 1. Given information.

Draw a segment with length $1$.

Step 2. Follow the ways.

To draw a segment of length $\sqrt{5}$, follow these simple ways-

Draw a segment with length of $1$.

Double it in the horizontal direction and take it as it is in vertical direction as shown in the figure.

Step 3. Complete the triangle.

Now complete the triangle, 

and the closing side will have length $\sqrt{5}$

since ($\because \sqrt{5}=\sqrt{2^2+1^2}$)

Step 4. Extend the length.

Now, extend the line of length $\sqrt{5}$, to 1 unit and draw a perpendicular bisector to find the midpoint,

 this half  length will be a required length.

Therefore, above figure is the required.

$\left(c\right)$

The figure is, 

Step 1. Given information.

Draw a segment with length $1$.

Step 2. Follow the ways.

To draw a segment of length $\sqrt{5}$, follow these simple ways-

Draw a segment with length of $1$.

Double it in the horizontal direction and take it as it is in vertical direction as shown in the figure.

Step 3. Complete the triangle.

Now complete the triangle, 

and the closing side will have length $\sqrt{5}$

since ($\because \sqrt{5}=\sqrt{2^2+1^2}$)

Step 4. Extend the length.

Now, extend the line of length $\sqrt{5}$, to 1 unit and draw a perpendicular bisector to find the midpoint,

 this half   length will be a required length.

Step 5. Draw the rectangle.

Now, from the marked half-length segment, and one unit length draw a rectangle

Therefore, the required figure drawn above.