construct an angle of 90° and angle of 45°at the initial point of a given ray write te steps of construction
please quick step by step answers needed
Answers
Answer:
1 .First, draw a ray OA with intial point O.
2.Taking O as a centre and some radius, draw an arc of a circle, which intersects OA, at a point B.
3.Taking B as centre and with the same radius as before, draw an arc intersecting the previous drawn arc, at a point C.
4.Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 2, say at D.
5.Draw the ray OE passing through C. Then ∠EOA = 60∘ .
6.Draw the ray OF passing through D. Then ∠ FOE = 60∘ .
7. Next, taking C and D as centres and with radius more than ½ CD, draw arcs to intersect each other, at G.
8. Draw the ray OG, which is the bisector of the angle FOE, i.e., ∠FOG = ∠EOG = 1/2 ∠FOE = 1/2 (60∘ ) = 30∘ .
Thus, ∠GOA = ∠GOE + ∠ EOA = 30∘ + 60∘ = 90∘.
9.Now taking O as centre and any radius more than OB, draw an arc to intersect the rays OA and OG, at H and I.
10. Next, taking H and I as centres and with the radius more than 1/2 HI, draw arcs to ntersect each other, at J.
11. Draw the ray OJ. This ray OJ is the required bisector of the ∠ GOA. Thus, ∠GOJ = ∠AOJ = 1/2 ∠GOA = 1/2(90∘) = 45∘.
Justification:
(i)Join BC.
Then, OC = OB = BC triangle. (By construction)
∴ ∠COB is an equilateral triangle.
∴ ∠COB = 60∘.
∴ ∠EOA = 60∘.
(ii)Join CD.
Then, OD = OC = CD (By construction)
∆DOC is an equilateral triangle.
∴ ∠DOC = 60∘.
∴ ∠ FOE = 60∘.
(iii)Join CG and DG.
In ΔODG and ΔOCG,
OD = OC[ Radii of the same arc]
DG=CG [Arcs of equal radii]
OG=OG [Common]
∴ Δ ODG = ΔOCG [SSS Rule]
∴ ∠ DOG= ∠COG [CPCT]
∴ ∠FOG = ∠ EOG = 1/2 ∠FOE = 1/2 (60∘) = 30∘
Thus, ∠GOA = ∠GOE + ∠EOA = 30∘ + 60∘ = 90∘
∴ ∠AOJ= ∠GOJ= 1/2 ∠GOA = ½(90°)=45
Answer:
Draw a line segment OA.
Taking O as center and any radius, draw an arc cutting OA at B.
Now, taking B as center and with the same radius as before, draw an arc intersecting the previously drawn arc at point C.
With C as center and the same radius, draw an arc cutting the arc at D.