Question

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Explanation:

Question 1

Step1: Understand Bearings

Bearings are measured clockwise from north (0°). Start with a north - south line (vertical) and east - west line (horizontal) as axes.

Step2: First Leg

For the first leg: bearing \(090^{\circ}\) (east direction) and distance \(56\)m. Draw a line from the origin (starting point) along the positive x - axis (east) with length proportional to \(56\)m.

Step3: Second Leg

The second bearing is between \(160^{\circ}\) and \(200^{\circ}\), which is in the third quadrant (south - west of the previous direction). Measure the bearing (e.g., if we take a mid - value like \(180^{\circ}\) for simplicity, but actually any value between \(160^{\circ}\) and \(200^{\circ}\)) from the positive y - axis (north) clockwise. From the end - point of the first line, draw a line at the given bearing (between \(160^{\circ}\) and \(200^{\circ}\)) with length proportional to \(35\)m.

Step4: Third Leg

The third bearing is \(310^{\circ}\), which is \(360 - 310=50^{\circ}\) west of north (or \(310^{\circ}\) clockwise from north). From the end - point of the second line, draw a line at \(310^{\circ}\) bearing with length proportional to \(42\)m.

Step5: Scale and Proportion

Use a suitable scale, e.g., \(1\)cm represents \(10\)m. So \(56\)m would be \(5.6\)cm, \(35\)m would be \(3.5\)cm, and \(42\)m would be \(4.2\)cm. Mark the starting point, then each subsequent point after each leg of the journey.

Step1: Analyze the Position of G and H

• Let's consider the coordinates. For triangle \(G\), let's assume the vertices are at \((-5,-3)\), \((-1,-3)\), \((-5,-5)\) (approximate from the graph). For triangle \(H\), the vertices are at \((-3,1)\), \((-3,5)\), \((-1,5)\) (approximate from the graph).

Step2: Vertical Translation

First, we can translate triangle \(G\) vertically. The vertical distance between the lowest point of \(G\) (y - coordinate \(-5\)) and the lowest point of \(H\) (y - coordinate \(1\)) is \(1-(-5)=6\) units. So we can translate \(G\) 6 units up.

Step3: Horizontal Translation

After vertical translation, we can translate horizontally. The horizontal distance between the left - most point of the vertically translated \(G\) (x - coordinate \(-5\)) and the left - most point of \(H\) (x - coordinate \(-3\)) is \(-3-(-5) = 2\) units. So we translate 2 units to the right.

Step4: Check Rotation (Optional)

We can also consider a rotation. The triangle \(G\) is in the fourth quadrant (if we consider the standard quadrants) and \(H\) is in the second quadrant. A rotation of \(90^{\circ}\) clockwise (or \(270^{\circ}\) counter - clockwise) can also be part of the transformation. Let's check the rotation:

• If we rotate triangle \(G\) \(90^{\circ}\) clockwise about the origin, the transformation of a point \((x,y)\) is \((y,-x)\). For a point \((-5,-3)\) in \(G\), after \(90^{\circ}\) clockwise rotation, it becomes \((-3,5)\), which is a vertex of \(H\). Then we can translate the rotated figure.

Answer:

1. Step 1: Set up the coordinate system

• Draw a vertical line (north - south) and a horizontal line (east - west) intersecting at a point (the starting point of the plane's journey). The vertical line represents the north - south direction (with north at the top) and the horizontal line represents the east - west direction (with east to the right).

1. Step 2: Draw the first leg of the journey

• A bearing of \(090^{\circ}\) means the plane is moving east (since bearings are measured clockwise from north). Using a scale (e.g., 1 cm = 10 m, so 56 m would be 5.6 cm), draw a line segment from the starting point along the horizontal (east - west) line towards the east (right - hand side) with a length corresponding to 56 m. Label the end - point of this segment as \(A\).

1. Step 3: Draw the second leg of the journey

• The second bearing is between \(160^{\circ}\) and \(200^{\circ}\). Bearings between \(180^{\circ}\) and \(270^{\circ}\) are in the south - west quadrant relative to the starting direction, but here we measure the bearing from the north (top of the vertical line) clockwise. From point \(A\), use a protractor to measure a bearing between \(160^{\circ}\) and \(200^{\circ}\) (e.g., if we choose \(180^{\circ}\) for simplicity, it would be directly south, but the actual direction is between south - east and south - west of the previous east - ward direction). Draw a line segment from point \(A\) at this bearing with a length corresponding to 35 m (3.5 cm with the 1 cm = 10 m scale). Label the end - point of this segment as \(B\).

1. Step 4: Draw the third leg of the journey

• A bearing of \(310^{\circ}\) is measured clockwise from north. Since \(310^{\circ}=360^{\circ}- 50^{\circ}\), it is \(50^{\circ}\) west of north. From point \(B\), use a protractor to measure a \(310^{\circ}\) bearing from the north (vertical) line. Draw a line segment from point \(B\) at this bearing with a length corresponding to 42 m (4.2 cm with the 1 cm = 10 m scale). Label the end - point of this segment as \(C\) (the final position of the plane).

1. Step 5: Label the diagram

• Label the starting point \(O\), the points \(A\), \(B\), and \(C\) as described. Also, label the bearings and the lengths of each leg of the journey. The key is to maintain the correct angular relationships (using a protractor for bearings) and proportional lengths (using a scale for distances) to accurately represent the plane's journey.

Question 2