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COLLEGE GEOMETRY – SAMPLE QUESTIONS
Contents
1.
2.
Elementary Geometry
Coordinate Geometry
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1. Elementary Geometry
1a. Two sides of a triangle are a = 8 and b = 6, the angle α (opposite to
the side a) is 45◦ . Calculate the missing side. Find the area of the triangle.
1b. Two sides of a triangle are a = 6 and b = 3, the angle α (opposite to
the side a) is 60◦ . Calculate the missing side and angles. Find the area of
the triangle.
1c. Two sides of a triangle and the angle enclosed by them are given: a=3,
b=4 and γ = 60◦ .
a) Find the missing side.
b) Calculate the area of the triangle.
c) Calculate the radius of the circumscribed circle.
√
1d. Three sides of a triangle are given: a=3, b=4 and c= 13.
a) Find the angle γ opposite to the side c.
b) Calculate the area of the triangle.
1e. The sides of a triangle are a = 5, b = 12 and c = 13. Calculate the angle
opposite to the side c.
1f. Three sides of a triangle are given: 13, 13 and 24.
a) Calculate the heights and the area of the triangle!
b) Calculate the biggest angle of the triangle!
c) Calculate the radius of the inscribed circle of the triangle!
2a. In a right triangle the longest side AB is 5. The leg BC is 3. Calculate
the missing leg. Find the radius of the circumscribed circle. Compute the
height belongig to the hypothenuse.
2b. The area of a right angle triangle is 30, the sum of the legs is 17.
Calculate the sides of the triangle.
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2c. The radius of the circumscribed circle around a right triangle is 5, one
of the legs is 6. What is the area of the triangle.
3a. In a symmetrical trapezoid the lenght of the shorter parallel base is 10.
The common lenght of the legs is 5 and the height is 4. What is the area of
the trapezoid?
3b. Three sides in a symmetrical trapezoid are of lenght 10, the fourth side
has lenght 20. Calculate the area of the trapezoid.
3c. In a symmetrical trapezium the inclination angle of the diagonal to the
longer parallel base is 45◦ , the lenght of the diagonal is 10. What is the area
of the trapezium?
3d. The longest base of a symmetrical trapezoid is 20, the length of the legs
is 5, the height is 3.
a) Calculate the area of the trapezoid!
b) Calculate the angles of the trapezoid !
3e. The longer diagonal of a rhombus is given: 12, and one of the angle of
the rhombus is 60◦
a) Calculate the area of the rhombus!
b) Calculate the length of the sides of the rhombus!
3f. The perimeter of a rhombus is 40, its area is 96. What are the angles,
sides, and diagonals of the rhombus?
4a. Calculate the common length of the sides of a regular 10-gon inscribed
in the unit circle.
4b. The length of the side of a regular 6-gon is given: 8.
a) Calculate the angles of the polygon!
b) What is the length of the shorter diagonal?
c) What is the area of the polygon?
4c. The lenght of the side of a regular 8-gon is given: 6. Calculate the area
of the polygon.
5a. Let a circle with radius 2 be given. The distance between the point P
and the center of the circle is 4. Calculate the common length of the tangent
segments from P to the given circle and find the length of the shorter arc
along the circle between the contact points A and B.
5b. Let a circle with radius 3 be given. The distance between the point P
and the center of the circle is 6. Calculate the common length of the tangent
segments from P to the given circle and find the length of the shorter arc
along the circle between the contact points A and B.
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5c. Two circles of radius 10 intersect each other. The distance of the their
centers is 16. Calculate the area of the common part of the two circles!
5d. Two circles of radius 5 intersect each other. The distance of the their
centers is 8. Calculate the area of the common part of the two circles!
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2. Coordinate Geometry
1a. Compute the distance between the points A(4, 1) and B(1, −1). Find
the coordinates of the midpoint of the segment AB.
1b. Let the points A(−3, 2), B(2, 3) and C(1, −2) be given. Compute the
angle at A in the triangle ABC. Find the area of the triangle.
Hint.
• compute the sides of the triangle,
• use the cosine rule
BC 2 = AB 2 + AC 2 − 2 · AB · AC · cos ∠A
to calculate the angle at A.
1c. Let the points A(−1, 1), B(1, 3) and C(3, 2) be given. Find the coordinates of the missing point D of the parallelogram ABCD. How many
solutions we have?
2a. Let the points A(4, 1), B(1, −1) and C(1, 2) be given. Find the equation
of the line
• passing through the points A and B,
• passing through the point C and parallel to the line AB,
• passing through the point C and perpendicular to the line AB.
Find the distance between the point C and the line AB.
2b. Find the equation of the line passing through the point C(3, 2) and
• parallel to y = −3x + 5
• perpendicular to y = 32 x + 46.
2c. Find the common point of the lines y = −2x + 3 and y = 3x − 8.
3a. Find the common points of the line y = 2x − 3 and the circle (x − 1)2 +
(y + 1)2 = 25.
3b. Find the equation of the tangent lines to the circle (x−1)2 +(y+1)2 = 16
which are parallel to y = 2x − 3.
Hint.
• determine the center of the circle: C(1, −1),
• write the equation of the line passing through C and perpendicular
to y = 2x − 3,
• the common points of the line in the previous step and the given
circle are the contact points of the tangent lines.
3c. Find the equation of the tangent line to the circle (x−1)2 +(y +1)2 = 25
at the point P (4, 3).
Hint.
• determine the center of the circle: C(1, −1),
COLLEGE GEOMETRY – SAMPLE QUESTIONS
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• write the equation of the line passing through C and P ,
• write the equation of the line passing through P and perpendicular
to the line CP .
4a. Find the common points of the circles (x + 1)2 + (y − 1)2 = 5 and
x2 + y 2 = 1
4b. Find the equation of the tangent lines to the circle (x−1)2 +(y+1)2 = 16
from the point P (3, 5).
Hint.
• determine the center of the circle: C(1, −1),
• calculate the coordinates of the midpoint of√
the segment CP : M (2, 2).
• calculate the distance between C and M : 10,
• the common points of the circles (x − 1)2 + (y + 1)2 = 16 and (x −
2)2 + (y − 2)2 = 10 will be the contact points of the tangent lines.
4c. Let the points A(−1, 1), B(1, 3) and C(3, 2) be given. Find the equation
of the circumscribed circle of the triangle ABC.
5. Explain why the following method is working well.
Excercise: find the common points of the circles (x − 1)2 + (y + 1)2 = 16
and (x − 2)2 + (y − 2)2 = 10.
Solution. (displacement-solution-displacement) We are going to find the
common points of the circles
(x + 1)2 + (y + 3)2 = 16
x2 + y 2 = 10.
Taking the difference we have
x + 3y = −2.
Therefore
(2 + 3y)2 + y 2 = 10
5y 2 + 6y − 3 = 0
√
√
−3 + 2 6
−3 − 2 6
y1 =
, y2 =
5
5
and, consequently,
x1 = −(2 + 3y1 ), x2 = −(2 + 3y2 ).
The solutions of the original problem are
A(x1 + 2, y1 + 2) and B(x2 + 2, y2 + 2)