Download 12.3 - Math TAMU

Section 12-3
Math 366 Lecture Notes
Section 12.3 – Other Constructions
A rhombus is a parallelogram in which all the sides are congruent.
Properties of a Rhombus:
1. A quadrilateral in which all the sides are congruent is a rhombus.
2. Each diagonal of a rhombus bisects the opposite angles.
3. The diagonals of a rhombus are perpendicular.
4. The diagonals of a rhombus bisect each other.
Constructing Parallel Lines
Rhombus method
Construct a line through P parallel to l.
P
l
1. Through P, draw any line that intersects l in A. PA will be a side of a rhombus.
2. Draw an arc with the pointer at A and radius AP to mark the third vertex, X, of a
rhombus.
3. With the same opening of the compass, draw intersecting arcs, first with the pointer at P
and then with the pointer at X to find Y, the fourth vertex of the rhombus.
4. Draw PY . PY // l in rhombus APYX.
Corresponding Angle method
Construct a line through P parallel to l.
P
l
1. Through P, draw a line that intersects l, forming angle α.
2. Copy angle α at point P. PY // l.
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Section 12-3
An angle bisector is a ray that separates an angle into two congruent angles.
Constructing Angle Bisectors
Construct an angle bisector of ∠ A.
A
1. With the pointer at A, draw any arc intersecting the angle at B and C, giving three vertices
of a rhombus with vertex at A.
2. Draw an arc with center at B and radius AB.
3. Draw an arc with center at C and radius AB. The arcs intersect at D, the fourth vertex of
the rhombus.
4. Connect A with D. AD is the angle bisector of A in rhombus ABDC. (The diagonals of a
rhombus bisect the angles.)
The distance from a point to a line is the length of the perpendicular segment from the point to
the line.
Constructing Perpendicular Lines
Construction of a Perpendicular to a Line through a Point not on the Line
Construct a line through P, perpendicular to line l (P is not a point on l).
P
l
1. Draw an arc with center at P that intersects the line at two points, A and B. (P, A, and B
will be vertices of a rhombus.)
2. With the same compass opening, make two intersecting arcs, one with center at A and the
other with center at B.
3. Connect P with Q, the intersection of the two arcs constructed in step 2. PQ is
perpendicular to l.
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Section 12-3
Construction of a Perpendicular to a Segment through the Midpoint of the Segment
This construction was done in section 12-1.
Construction of a Perpendicular to a Segment through a Point on the Segment
Construct a line through P, perpendicular to line l (P is a point on l).
P
l
1. Draw an arc with center at P that intersects l in two points, A and B. ( AB will be the
diagonal of a rhombus.)
2. Use a larger opening for the compass and draw intersecting arcs, with centers at A and B,
to determine C and D (endpoints of the other diagonal of the rhombus).
3. Connect C with D, the points where the arcs intersect. CD is perpendicular to l through
P. ( CD and AB are perpendicular bisectors of each other in rhombus ACBD.)
Properties of Angle Bisectors
Theorem 12-8
a. Any point P on an angle bisector is equidistance from the sides of the angle.
b. Any point that is equidistant from the sides of an angle is on the angle bisector of the
angle.
Proof of part a:
Given: AP is the angle bisector of ∠ BAC.
Prove: P is equidistant from AB and AC .
B
A
P
C
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Section 12-3
Theorem 12-9
The angle bisectors of a triangle are concurrent and the three distances from the point of
intersection to the sides are equal.
A line is tangent to a circle if it intersects the circle in one and only one point and is
perpendicular to a radius.
A circle is inscribed in a triangle if all the sides of the triangle are tangent to the circle. The
inscribed circle is the incircle; the center is the incenter.
Constructing a Circle Inscribed in a Triangle.
Inscribe a circle in ∆ ABC.
B
A
C
1. Bisect the angles of the triangles. The intersection of the angle bisectors, P, will be the
center of the circle.
2. Construct a perpendicular from P to a side of the triangle. The length of that segment
will be the length of the radius of the circle.
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