Art Department Unit 2: Tangents to circles. 3º ESO TECHNICAL DRAWING
UNIT 2: TANGENTS TO CIRCLES
Art Department Unit 2: Tangents to circles. 3º ESO 1. TANGENTS TO CIRCLES
1.1. Tangents through an external point
1.2. Common tangents
a. To draw common external tangents to two unequal circles
b. To draw common internal tangents to two unequal circles.
1.3. To draw circles with radius r2 tangent to a given line and a given
circle
1.4. To draw a circle of given radius, r, tangent to two given lines s and t
1.5. To draw a circle of radius r tangent to two given circles
1. TANGENTS TO CIRCLES
-
Tangent to a circle, is a line in the plane of the circle that intersects the
circle in exactly one point. It is perpendicular to the radius drawn to the
point of tangency.
The blue line in the figure is called the "tangent to the circle c". Another way
of saying it is that the blue line is "tangential" to the circle c.
The line barely touches the circle at a single point. If the line were closer to
the center of the circle, it would cut the circle in two places and would then
be called a secant. In fact, you can think of the tangent as the limit case of a
secant. As the secant line moves away from the center of the circle, the two
points where it cuts the circle eventually merge into one and the line is then
the tangent to the circle.
As can be seen in the figure, the tangent line is always at right angles to the
radius at the point of contact.
Page 2 of 9 Art Department Unit 2: Tangents to circles. 3º ESO 1.1.
Tangents through an external point
We start with a given circle with center C, and a point P outside the
circle.
- Draw a straight line between the center
O of the given circle and the given point
P.
- Find the midpoint of this line by
constructing the line's perpendicular
bisector. The midpoint may be inside or
outside the circle, depending on the
circle size and the location of the given
point.
- Place the compass on the midpoint just
constructed, and set it's width to the
center C of the circle.
- Without changing the width, draw an arc
across the circle in the two possible
places. These are the contact points J,
K for the tangents.
- Draw the two tangent lines from P
through J and K. The two lines PJ and
PK just drawn are tangential to the
given circle and pass through P.
Page 3 of 9 Art Department Unit 2: Tangents to circles. 3º ESO 1.2.
Common tangents
A common tangent is a line that is tangent to each of two coplanar circles. A
common tangent can be tangent either internally or externally:
- A common internal tangent is a common tangent that intersects the
segment that joins the center of the two circles.
- A common external tangent is a common tangent that does not
intersect the segment that joins the centers of the two circles.
a.
To draw common external tangents to two unequal circles
Step 1: Join the centres O and O’ of the circles. Mark OM equal to MO’.
With centre M and radius MO draw a circle.
Step 2: With centre O’ and radius r’ – r draw a circle. This circle cuts the
previous circle with centre M of step 1 at 1 and 2.
Draw a line from O’ through 1 cutting the circumference of the large
circle at T’1.
Draw a line from O’ through 2 cutting the circumference of the large
circle at T’2.
Step 3: Draw OT1 parallel to O’T’1. Draw OT2 parallel to O’T’2.
Step 4: Draw a line through T1 and T’1 → t1.
Draw a line through T2 and T’2 → t2.
These are the requiered tangents.
Step 1
Step 2
Step 3
Paso 4
Page 4 of 9 Art Department Unit 2: Tangents to circles. 3º ESO Step 4: Solution
b.
To draw common internal tangents to two unequal circles.
Step 1: Join the centres O and O’ of the circles A. Mark OM equal to
MO’. With centre M and radius MO draw a circle.
Step 2: With centre O’ and radius r’ + r draw a circle. This circle cuts
the previous circle with centre M of step 1 at 1 and 2.
Draw a line from O’ to1 cutting the circumference of the circle
with centre O’ and radius r’ at T’1.
Draw a line from O’ to 2 cutting the circumference of the circle
with centre O’ and radius r’ at T’2.
Step 3: Draw OT1 parallel to O’T’1. Draw OT2 parallel to O’T’2.
Step 4: Draw a line through T1 and T’1 → t1.
Draw a line through T2 and T’2 → t2.
These are the requiered tangents.
Page 5 of 9 Art Department Unit 2: Tangents to circles. 3º ESO Step 1
Step 2
Step 3
Paso 4
Step 4: Solution
Page 6 of 9 Art Department Unit 2: Tangents to circles. 3º ESO 1.3.
To draw circles with radius r2 tangent to a given line and a given circle
Step 1:
Step 2:
Draw a parallel at a distance r2 from the given line s.
Add the given radius r2 to the radius r of the given circle (r+r2)
and draw an arc.
The crossing O1 and O2 are the centres of the required circles.
Draw the circles with centre O1 and O2 and radius r2. These
circles will tangent the given line s and circle.
Step 1
Paso 2
Step 2
Paso 4
Page 7 of 9 Art Department Unit 2: Tangents to circles. 3º ESO 1.4.
To draw a circle of given radius, r, tangent to two given lines s and t.
Step 1: Bisect the acute angle formed by the two given lines t and s.
Step 2: Draw a parallel line u at a distance r from the given line t. This line
cuts the angle bisector of step 1 at point O.
Step 3: Use the compass to construct a circle with center at O and radius
of length OT1 and OT2, which are of equal length. Since OT1 and
OT2 are perpendicular to t and s, the circle constructed from them
is tangent to t and s.
Step 1
Step 2
Step 3
Paso 4
Page 8 of 9 Art Department Unit 2: Tangents to circles. 3º ESO 1.5.
To draw a circle of radius r tangent to two given circles
Step 1:
Step 2:
Step 3:
Step 1
Draw the two circles of given radius r1 and r2.
With centre O1 and radius r1 + r draw a circle.
With centre O2 and radius r2 + r draw a circle.
The intersections of these circles are the required centres O3 and
O4.
Draw two circles of radius r and centre O3 and O4 respectively.
Step 2
Step 3
Paso 4
Page 9 of 9