Using a ruler and a per of compasses only construct triangle ABC such that AB=10cm and angle BAC=45° and angle ABC=30°
draw AB
construct ∡A (bisect a right angle there)
construct ∡B (bisect a 60° angle there)
Where rays from A and B intersect is point C.
Answer for it
To construct triangle ABC with AB = 10 cm and ∠BAC = 45° and ∠ABC = 30° using a ruler and a pair of compasses, follow the steps below:
Step 1: Draw a line segment XY of any arbitrary length using the ruler.
Step 2: Place the compass on point X and extend it to point Y.
Step 3: Without changing the radius, place the compass on point Y and draw an arc cutting the line segment XY. Label the point where the arc intersects XY as point Z.
Step 4: Set the compass to a radius of 10 cm and place the compass on point Z.
Step 5: Draw an arc above line segment XY that intersects the arc drawn in step 3. Label the point of intersection as point A.
Step 6: With the compass still set at a radius of 10 cm, place the compass on point A.
Step 7: Draw an arc that intersects the previous arc drawn in step 5. Label this point of intersection as point B.
Step 8: With the compass set to a radius greater than half the length of line segment AB, place the compass on point B.
Step 9: Draw an arc that intersects the arc drawn in step 5. This will create two points of intersection, label the lower point as point C.
Step 10: Draw the line segments AB and BC to complete triangle ABC.
Now, you have successfully constructed triangle ABC with AB = 10 cm, ∠BAC = 45°, and ∠ABC = 30° using only a ruler and a pair of compasses.
To construct triangle ABC with AB = 10 cm, ∠BAC = 45°, and ∠ABC = 30° using a ruler and a pair of compasses, follow these steps:
1. Draw a line segment AB of length 10 cm using a ruler.
2. Place the sharp end of the compass at point A and draw an arc that intersects line AB.
3. Without changing the compass width, place the sharp end of the compass at the intersection of the arc and line AB and draw another arc.
4. Label the intersection of the second arc and the original arc as point C.
5. Place the sharp end of the compass at point B and draw an arc that intersects line AB.
6. Without changing the compass width, place the sharp end of the compass at the intersection of the arc and line AB and draw another arc.
7. Label the intersection of the second arc and the original arc as point D.
8. Using a ruler, draw a line segment CD connecting points C and D.
9. Now, you have the triangle ABC with AB = 10 cm. To determine the angles, you'll need some additional constructions:
a. Place the sharp end of the compass at point B and draw an arc that intersects line BC.
b. Without changing the compass width, place the sharp end of the compass at the intersection of the arc and line BC and draw another arc.
c. Label the intersection of the second arc and the original arc as point E.
d. Draw a line segment AE using a ruler.
e. In the quadrilateral BCED, you can see that ∠BEC = 2 × ∠BAC as the angle at the center is twice the angle at the circumference.
f. Using your compass, set its width to BO (equal to AC) and place its sharp end at point B. Draw an arc that intersects line AE.
g. Without changing the compass width, place the sharp end of the compass at the intersection of the arc and line AE and draw another arc.
h. Label the intersection of the second arc and the original arc as point F.
i. Draw a line segment BF using a ruler.
10. Now, you have triangle EBF with known angles. To construct the angle ∠ABC = 30°, follow these steps:
a. Place the sharp end of the compass at point B and draw an arc within the triangle EBF.
b. Without changing the compass width, place the sharp end of the compass at point F and draw another arc that intersects the previous arc.
c. Label the intersection as point G.
d. Draw a line segment BG using a ruler.
11. Triangle ABC is complete, and you have successfully constructed it with the given specifications.
Remember to use a ruler for drawing straight lines and a compass for measuring and drawing arcs. Take your time and be precise in your constructions to get accurate results.