Exercises Set 4.2
1.a.
Construct a line segment AB of given length 4d and mark its midpoint, M. Construct a circle of
radius 2d centered at point A and passing through point M. Construct a line segment AM and
mark its midpoint, T. Construct a circle of radius 3d centered at point B and passing through point
T. Mark one of the points of intersection of the two circles point C. Triangle ABC is a scalene
triangle with sides 2d, 3d, and 4d.
1.b.
Construct a line segment AB of given length 2d. Construct a line perpendicular to segment AB
and passing through point A labeled k. Construct a circle of radius 2d centered at point A and
passing through point B. Mark one of the points of intersection of this circle and the line k point
D. Construct a line segment AD and mark its midpoint M. Construct a circle of radius d centered
at point D and passing through point M. Mark the point of intersection of this smaller circle and
line k point C. Triangle ABC is a right triangle with sides 2d and 3d.
1.c.
Construct a line segment AB of given length d. Construct a circle centered at point B and passing
through point A. Label the point of intersection of this circle with line AB point C. Construct
another circle centered at point C and passing through point B. Label the point of intersection of
this circle and the line AB point D. Note, point D is a distance 3d from point A. Construct a line
perpendicular to line segment AB and passing through point A. Repeat the process of constructing
circles of radius d along this perpendicular line until a point, G, is established at a distance 4d from
point A. Triangle ADG is a right triangle with sides 3d and 4d and hypotenuse 5d.
1.d.
Construct a line segment AD of given length 2d and mark its midpoint, M. Construct a circle of
radius d centered at point D and passing through point M. Label the point of intersection of this
circle and the line AD point B. Note, point B is a distance 3d from point A. Construct a circle of
radius 3d centered at point B and passing through point A. Construct a circle of radius 2d centered
at point A and passing through point D. Mark one of the points of intersection of the last two
constructions point C. Triangle ABC is an isosceles triangle with sides 2d, 3d, and 3d.
1.e.
Construct a line segment AB of given length 3d. Construct a circle of radius 3d centered at point
B and passing through point A. Construct a circle of radius 3d centered at point A and passing
through point B. Mark one of the points of intersection of the last two constructions point C.
Note, point C is a distance 3d from both points A and B. Triangle ABC is an equilateral triangle
with sides 3d.
2.a.
never true
2.b.
sometimes true
2.c.
sometimes true
2.d.
always true
2.e.
always true
2.f.
sometimes true
2.g.
sometimes true
2.h.
never true
2.i.
sometimes true
3.
Circles R and S overlap some. Circle E has nothing in common with either circle R or circle S.
All three circles are inside the rectangle U.
U
R
4.
S
E
Circles R and A overlap some. Circles A and E overlap some. Circle O has nothing in common
with any of the other circles. All three circles are inside the rectangle U.
U
R
5.
A
E
O
Circles O and I overlap some. Circles I and A overlap some. Circle E overlaps both circle A and
circle I. All four circles are inside the rectangle U.
U
O
I
A
E
6.a.
The circumcircle of the triangle is a circle with center O passing through all three vertices of the
triangle. The circumcenter O is the point of intersection of the perpendicular bisectors.
6.b.
The incircle of the triangle is a circle with center P that meets each side of the triangle in only one
point that is not a vertex. The incenter P is the point of intersection of the angle bisectors.
6.c.
The Euler Line is a line that contains the centroid, G, the orthocenter, H, and the circumcenter, O.
6.d.
The centroid, G, lies two-thirds of the way between the orthocenter, H, and the circumcenter, O,
except in the case of the equilateral triangle where G, H, and O are all the same point within the
equilateral triangle.
7.a.
It is not possible for the centroid, G, to lie on a vertex of the triangle because it is located at the
intersection of the medians in the triangle.
7.b.
It is not possible for the centroid, G, to lie on a side of the triangle because it is located at the
intersection of the medians in the triangle.
7.c.
It is not possible for the centroid, G, to lie outside the triangle because it is located at the
intersection of the medians in the triangle.
8.a.
It is possible for the orthocenter, H, to lie on a vertex of the triangle when the triangle is a right
triangle. Actually, H will be located at the vertex that measures 90°.
8.b.
It is possible for the orthocenter, H, to lie on a side of the triangle when the triangle is a right
triangle. Actually, H will be located where the two sides meet to form a 90° angle.
8.c.
It is possible for the orthocenter, H, to lie outside the triangle when the triangle is an obtuse
triangle.
9.a.
It is not possible for the circumcenter, O, to lie on a vertex of the triangle because by definition it
is the center of the circle which passes through all three vertices of the triangle.
9.b.
It is possible for the circumcenter, O, to lie on a side of the triangle when the triangle is a right
triangle. Actually, O will be located on the hypotenuse of the right triangle.
9.c.
It is possible for the circumcenter, O, to lie outside the triangle when the triangle is an obtuse
triangle.
10.a.
It is not possible for the incenter, P, to lie on a vertex of the triangle because by definition it is the
center of the circle that meets each side of the triangle in only one point that is not a vertex.
10.b.
It is not possible for the incenter, P, to lie on a side of the triangle because by definition it is the
center of the circle that meets each side of the triangle in only one point that is not a vertex.
10.c.
It is not possible for the incenter, P, to lie outside the triangle because by definition it is the center
of the circle that meets each side of the triangle in only one point that is not a vertex.
11.
The values of these ratios are equivalent to one another. The corresponding angles in triangles
ADE and ABC are congruent.
12.
One method of calculating the area of the given trapezoid is to use the formula A = ½ (sum of the
bases) (altitude). This yields the result A = ½ (a² + b²) + ab. Another method for calculating the
area of the given trapezoid is to combine the areas for the three triangles which are enclosed by the
trapezoid. Recall the formula for the area of a triangle is A = ½ (base) (altitude). This method
yields the result A = ab + ½c². Setting the results of the two methods equal to each other since
they both describe the area of the same trapezoid and simplifying yields the equation a² + b² = c².