Constructions
What tool do we use to copy a length or make a circle in a construction?
A compass!
What type of triangle always has at least two equal sides?
Isosceles triangle.
Define a circle.
A circle is the set of all points equidistant from a given point (the center).
True or False: The perpendicular bisector of AB always passes through the midpoint of AB.
True
You construct two circles with the same center but different radii. What is this figure called?
Concentric circles.
If two circles intersect at points X and Y, what can you say about line XY?
Line XY passes through both intersections and is the perpendicular bisector of the segment connecting the centers
If quadrilateral ABCD has all four sides equal, what kind of shape could it be?
Could be a rhombus or a square.
Define perpendicular bisector.
A line through the midpoint of a segment that splits the midpoint into two congruent halves.
If point P lies on the perpendicular bisector of AB, what can you say about distances PA and PB?
PA = PB (equal distances from A and B).
How can you prove two triangles are congruent using a compass without measuring with a ruler?
Show that each line segment has the same radii (SSS)
Why does constructing a hexagon inside a circle require using the circle’s radius?
All points on the bisector are equidistant from A and B.
A quadrilateral is constructed using two intersecting circles with equal radii. What figure results?
A rhombus
Define congruent segments
Segments with the same length.
Triangle XYZ is isosceles with XY = XZ. Which two angles must be congruent?
∠ Y = ∠ Z (base angles opposite equal sides are congruent)
Why does constructing a hexagon inside a circle require using the circle’s radius?
Because the distance around a circle can be divided into 6 equal parts by its radius. If you use the radius to “step” around the circle, you land on 6 equally spaced points. Connecting those points makes a regular hexagon
When constructing an equilateral triangle with circles, why are all sides guaranteed equal?
Circles centered at each endpoint with radius equal to the segment ensure all three sides are equal.
What’s the difference between a rhombus and a square?
A rhombus has all sides equal but angles not necessarily 90°; a square has all sides equal and all right angles.
If a triangle is equilateral, what can you conclude about its angles?
A polygon with all sides and angles equal.
A student says: “If two lines are perpendicular to the same line, then they must be parallel.” Is this always true?
Yes, always true (if two distinct lines are perpendicular to the same line, they must be parallel to each other).
If a triangle is equilateral, what can you conclude about its angles?
Each angle = 60°.
Draw a circle.
Construct a diameter.
Construct a perpendicular diameter.
Connect the 4 points of intersection with the circle
What shape do you get?
Square
Explain why opposite sides of a parallelogram are congruent using construction reasoning.
By construction, each pair of opposite sides is drawn with equal radii from intersecting circles, so they are congruent.
Define conjecture in geometry.
A statement believed to be true based on observations, but not yet proven.
Explain why the diagonals of a square are perpendicular and equal.
In a square, diagonals are equal (sides are congruent and right angles enforce equality) and diagonals meet at right angles (by symmetry).
A student tries to construct a regular hexagon but ends up with unequal sides. What mistake might they have made?
Likely error: Did not use the circle’s radius to step around the circle correctly (sides become uneven).