1. To bisect an angle using a ruler and compass, you should draw:
a) An arc with the angle's vertex as the center
b) A circle with the angle's vertex as the center
c) A straight line from the angle's vertex
d) Two arcs from the sides of the angle intersecting each other
Answer:
d) Two arcs from the sides of the angle intersecting each other
Explanation:
By drawing arcs from the sides of the angle which intersect, you find the midpoint which helps in bisecting the angle.
2. To construct a perpendicular bisector to a line segment, the arcs in the construction should be drawn with:
a) The segment's endpoints as centers
b) The midpoint of the segment as the center
c) Any point on the segment as the center
d) The intersection point of the segment and the circle as the center
Answer:
a) The segment's endpoints as centers
Explanation:
By drawing arcs from both endpoints of the segment that intersect, the perpendicular bisector can be constructed.
3. When constructing a triangle given its three sides (SSS), you first:
a) Draw the base of the triangle
b) Draw the height of the triangle
c) Bisect the largest side
d) Draw the circumcircle of the triangle
Answer:
a) Draw the base of the triangle
Explanation:
You start by drawing one side, then use compass measurements to get the other two sides from the endpoints.
4. If you're constructing a triangle given two sides and the included angle (SAS), where do you draw the first arc?
a) From the vertex of the angle
b) From one endpoint of the base
c) From both endpoints of the base
d) From the midpoint of the base
Answer:
a) From the vertex of the angle
Explanation:
The arc will determine where the third vertex of the triangle will lie.
5. When constructing an angle of 60° using a compass, you essentially construct:
a) An equilateral triangle
b) A square
c) A rectangle
d) A right triangle
Answer:
a) An equilateral triangle
Explanation:
When you draw an equilateral triangle, all its angles are 60°.
6. The construction of a right angle using only a straightedge and compass is equivalent to:
a) Halving an angle of 45°
b) Bisecting an angle of 90°
c) Tripling an angle of 30°
d) Quadrupling an angle of 22.5°
Answer:
b) Bisecting an angle of 90°
Explanation:
When you bisect a straight angle (180°), you get a right angle.
7. Which of the following cannot be constructed using just a straightedge and compass?
a) Angle of 20°
b) Angle of 45°
c) Angle of 60°
d) Angle of 90°
Answer:
a) Angle of 20°
Explanation:
Some angles, like the 20°, cannot be trisected using only a straightedge and compass.
8. In a triangle ABC, if the internal bisector of angle A meets side BC at point D, then the line AD:
a) Is the altitude of ABC
b) Is the median of ABC
c) Divides BC in the ratio of AB to AC
d) Is perpendicular to BC
Answer:
c) Divides BC in the ratio of AB to AC
Explanation:
This is a property of the internal angle bisector of a triangle.
9. To construct a tangent to a circle from an external point P, you first:
a) Draw the radius to point P
b) Find the midpoint of segment joining the center of the circle to P
c) Construct a perpendicular bisector to segment joining the center of the circle to P
d) Construct an angle of 45° with point P
Answer:
b) Find the midpoint of segment joining the center of the circle to P
Explanation:
The tangent at the point where the circle and the line from P to its midpoint intersect is the required tangent.
10. To construct an angle of 45°, you first construct:
a) An angle of 90°
b) An equilateral triangle
c) An angle of 60° and then bisect it
d) A square and then measure one of its angles
Answer:
a) An angle of 90°
Explanation:
Once you've constructed a 90° angle, bisecting it will yield a 45° angle.
11. In a triangle ABC, constructing the orthocenter involves finding:
a) The intersection of the medians
b) The intersection of the altitudes
c) The intersection of the angle bisectors
d) The circumcenter of the triangle
Answer:
b) The intersection of the altitudes
Explanation:
The orthocenter is the point where all the altitudes of a triangle intersect.
12. In a triangle ABC, constructing the circumcenter involves:
a) Drawing the perpendicular bisectors of the sides
b) Drawing the angle bisectors
c) Drawing the medians
d) Drawing the altitudes
Answer:
a) Drawing the perpendicular bisectors of the sides
Explanation:
The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect.
13. To construct the bisector of an angle using a compass and straightedge, you should:
a) Use only the straightedge
b) Use only the compass
c) Use both the compass and the straightedge
d) Neither the compass nor the straightedge
Answer:
c) Use both the compass and the straightedge
Explanation:
You'll need to use the compass to draw arcs and the straightedge to draw the bisector line.
14. To duplicate a given angle, the tool necessary is:
a) Only a straightedge
b) Only a compass
c) Both a straightedge and a compass
d) Neither a straightedge nor a compass
Answer:
c) Both a straightedge and a compass
Explanation:
To duplicate an angle, you'll need to use the compass to transfer lengths and the straightedge to draw lines.
15. To trisect a line segment means to:
a) Divide it into three equal parts
b) Make three copies of it
c) Cut it into three pieces of any length
d) Measure its length three times
Answer:
a) Divide it into three equal parts
Explanation:
Trisecting means dividing something into three equal parts.
16. To construct the perpendicular bisector of a given line segment, the necessary tools are:
a) Only a straightedge
b) Only a compass
c) Both a straightedge and a compass
d) A protractor
Answer:
c) Both a straightedge and a compass
Explanation:
The compass is used to mark arcs and the straightedge is used to draw the bisector.
17. Which is NOT a basic constructible polygon using only a straightedge and compass?
a) An equilateral triangle
b) A square
c) A pentagon
d) A heptagon (7-sided polygon)
Answer:
d) A heptagon (7-sided polygon)
Explanation:
While certain heptagons can be constructed, a regular heptagon cannot be constructed using only a straightedge and compass.
18. The point where the medians of a triangle intersect is called the:
a) Circumcenter
b) Incenter
c) Orthocenter
d) Centroid
Answer:
d) Centroid
Explanation:
The centroid is the point of concurrency of the three medians of a triangle.
19. If you need to draw a circle touching all the sides of a triangle, you would be constructing:
a) Circumcircle
b) Incircle
c) Excircle
d) Semicircle
Answer:
b) Incircle
Explanation:
An incircle touches all the sides of a triangle from inside.
20. The construction which requires just a straightedge (and not a compass) is:
a) Bisecting an angle
b) Drawing a perpendicular from a point on a line
c) Drawing a tangent to a given circle
d) Constructing the midpoint of a segment
Answer:
b) Drawing a perpendicular from a point on a line
Explanation:
Given a point on a line, one can use a straightedge to draw a perpendicular by aligning it appropriately with the given line.