Unit 5.1: Measuring, arealand shape. Draw a perpendicular line from a point onto a straight line and then draw an angle of \( 60^{\circ} \) on the straight line measured from the perpendicular line by using a ruler and compass only.

Certainly! Let's walk through the step-by-step process to: 1. **Draw a perpendicular line from a given point to a straight line.** 2. **Construct a \(60^\circ\) angle on the straight line measured from the perpendicular line.** We'll use only a **compass** and a **straightedge (ruler without markings)** for this construction. --- ## **Given:** - A straight line \( \ell \). - A point \( P \) not on line \( \ell \). ## **Objective:** 1. **Construct the perpendicular from point \( P \) to line \( \ell \).** 2. **At the foot of the perpendicular, create a \(60^\circ\) angle between the perpendicular line and line \( \ell \).** --- ## **Materials Needed:** - Compass - Straightedge (ruler without markings) --- ## **Step 1: Constructing the Perpendicular from Point \( P \) to Line \( \ell \)** 1. **Locate the Given Elements:** - Draw the straight line \( \ell \) horizontally for ease of construction. - Mark the point \( P \) somewhere above or below line \( \ell \). 2. **Draw Arcs from Point \( P \) to Intersect Line \( \ell \):** - Place the compass tip on point \( P \). - Open the compass to a width that allows you to draw two arcs intersecting line \( \ell \). The width should be more than half the distance between \( P \) and \( \ell \) to ensure two intersection points. - Draw an arc below or above \( P \) that intersects line \( \ell \) at two points. Label these intersection points as \( A \) and \( B \).  *(Placeholder for illustration)* 3. **Find the Midpoint between Points \( A \) and \( B \):** - **Construct Arcs Above and Below Line \( \ell \):** - Without changing the compass width, place the compass on point \( A \) and draw an arc above and below line \( \ell \). - Repeat the same from point \( B \). The two sets of arcs will intersect above and below the line. - Label the points where the arcs intersect as \( C \) and \( D \).  *(Placeholder for illustration)* 4. **Draw the Perpendicular Bisector:** - Use the straightedge to draw a straight line connecting points \( C \) and \( D \). - This line will intersect line \( \ell \) at point \( F \), which is the foot of the perpendicular from \( P \) to \( \ell \). - The line \( PF \) is the **perpendicular** from \( P \) to \( \ell \).  *(Placeholder for illustration)* --- ## **Step 2: Constructing a \(60^\circ\) Angle at Point \( F \) on Line \( \ell \)** 1. **Prepare to Construct the Angle:** - You now have the perpendicular line \( PF \) intersecting line \( \ell \) at point \( F \). - The objective is to create a \(60^\circ\) angle between \( PF \) and \( \ell \) at \( F \). 2. **Drawing an Arc to Mark Reference Points:** - Place the compass tip on point \( F \). - Open the compass to a moderate width and draw an arc that intersects both lines \( PF \) and \( \ell \). - Label the intersection points as \( G \) on \( PF \) and \( H \) on \( \ell \).  *(Placeholder for illustration)* 3. **Constructing an Equilateral Triangle to Achieve \(60^\circ\):** - **With the same compass width, draw an arc from point \( H \):** - Place the compass tip on point \( H \) and draw an arc above line \( \ell \). - **Intersect the Arc with the First Arc:** - Without changing the compass width, place the compass tip on point \( G \) and draw another arc that intersects the arc drawn from \( H \). - Label the intersection point as \( I \).  *(Placeholder for illustration)* 4. **Drawing the \(60^\circ\) Angle:** - Use the straightedge to draw a straight line from point \( F \) to point \( I \). - The angle \( \angle IFP \) formed between \( FI \) and \( PF \) is \(60^\circ\).  *(Placeholder for illustration)* --- ## **Final Result:** - You now have: - A **perpendicular line** \( PF \) from point \( P \) to line \( \ell \). - A \(60^\circ\) **angle** between the perpendicular line \( PF \) and line \( \ell \) at the foot \( F \). --- ## **Visual Summary:** While I can't provide actual images, here's a textual representation of the final configuration: ``` P | | | FI | / | / | / ) F----------H ``` - **\( PF \)** is the perpendicular from \( P \) to \( \ell \). - **\( FI \)** creates a \(60^\circ\) angle with \( PF \) at point \( F \). - **\( \ell \)** is the original straight line. --- ## **Tips:** - **Accuracy with Compass:** Ensure your compass remains at the same width when required during the construction to maintain precision. - **Steady Hand with Straightedge:** Use the straightedge to draw clean, straight lines without bending or shifting to keep the construction accurate. - **Double-Check Intersections:** Verify the intersection points of arcs carefully to avoid errors in the final angle. By following these steps meticulously, you can successfully perform the desired geometric constructions using only a compass and a straightedge.