Using a ruler and a pair of compasses only construct angle ABC such that[AB]=10cm,angle ABC=30°and angle BAC=45°.A perpendicular from C to AB at D.Measure[CD]
Step 1: Draw a line segment AB of length 10 cm using the ruler.
Step 2: Draw a ray AC at an angle of 45 degrees to AB using the compasses. Place the tip of the compasses at point A and draw an arc that intersects AB. Without changing the width of the compasses, place the tip at the intersection point and draw another arc that intersects the first arc. This creates an angle of 45 degrees.
Step 3: Draw a ray BD at an angle of 30 degrees to AB using the compasses. Place the tip of the compasses at point B and draw an arc that intersects AB. Without changing the width of the compasses, place the tip at the intersection point and draw another arc that intersects the first arc. Draw a ray from B that passes through the intersection point and label it as BD.
Step 4: Construct a perpendicular from point C to AB. Place the tip of the compasses at point C and draw an arc that intersects AC and BD. Without changing the width of the compasses, place the tip at the intersection point and draw another arc that intersects the first arc. Draw a line segment from C to the intersection point on AB, label the intersection point as D.
Step 5: Measure CD using the ruler.
To construct the angle ABC using a ruler and a pair of compasses, follow these steps:
1. Draw a straight line segment AB of length 10 cm using a ruler.
2. Place the compass at point A and draw an arc with any radius greater than half of AB. Label the intersection points of the arc with the line as E and F.
3. Without changing the compass width, place the compass at point E and draw another arc intersecting the previous arc at point G.
4. With the same compass width, place the compass at point F and draw another arc intersecting the previous arc at point H.
5. Draw a straight line segment GH using a ruler.
6. Place the compass at point G and adjust the width to the distance between G and H.
7. Draw an arc that intersects the line segment GH at point I.
8. Without changing the compass width, place the compass at point H and draw another arc intersecting the previous arc at point J.
9. Draw a straight line segment IJ using a ruler.
10. The angle formed by lines AB and IJ is angle ABC.
To measure the length of CD (the perpendicular from C to AB), follow these steps:
1. Extend line segment AB beyond point B using a ruler.
2. Without changing the compass width, place the compass at point C and draw an arc that intersects the extended line at point K.
3. Draw a straight line segment CK using a ruler.
4. Measure the length of CD using a ruler, which is the distance between points C and K.
To construct angle ABC, follow these steps using a ruler and a pair of compasses:
1. Draw a line segment AB of length 10 cm. This can be done by using the ruler to measure the desired length and drawing a straight line between the points A and B.
2. Place the compass at point A and draw an arc that intersects the line AB. This arc should have a radius greater than half of AB, but not too large so as to intersect any other constructions.
3. Without changing the compass width, place the compass at the intersection point of the arc and line AB (let's call it E).
4. Draw another arc from point E, which intersects the previous arc. Let the intersection point be F.
5. Using the ruler, draw a straight line from point A to point F. This line represents AF.
6. Now, let's construct the angle BAC. Place the compass at point A, and draw an arc that cuts the AF line at point G.
7. Without changing the compass width, place the compass at point G and draw an arc intersecting the previous arc. Let the intersection point be H.
8. Using the ruler, draw a straight line from point A to point H. This line represents AH.
9. Now, let's construct the angle ABC of 30°. Place the compass at point B and draw an arc that cuts the line AH at point I.
10. Without changing the compass width, place the compass at point I and draw an arc intersecting the previous arc. Let the intersection point be J.
11. Using the ruler, draw a straight line from point B to point J. This line represents BJ.
12. The intersection point of lines BJ and AF is point C.
To measure [CD], follow these steps:
13. Using the ruler, draw a line perpendicular to AB from point C. Let the line intersect AB at point D.
14. Using the ruler, measure the length of line segment CD to find the desired measurement.
Remember to use the ruler to measure and draw straight lines accurately, and adjust the compass width carefully to ensure precise constructions.