First, we need to classify the quadrilateral according to the given properties. To do so we will make a table containing all special quadrilaterals and we will consider whether these quadrilaterals satisfy the properties.
A rectangle meets all of the properties, so ABCD is a rectangle. Now, let's construct a rectangle. We can begin by drawing the segment AC with the help of a straight edge.
We want to draw BD, which bisects AC and is not perpendicular to AC. From here we will find the perpendicular bisector of AC using a compass. Then we will draw BD, which is not on the perpendicular bisector. Let's put the compass on point A, and draw an arc that passes through point C.
We will use the same compass settings and draw another arc by putting the compass on point C. After that, we will label the intersection points of arcs as E and F.
Now we will draw EF, which is the perpendicular bisector of AC. We know that EF bisects AC at its midpoint, M.
From here, we will draw a circle whose center is at point M and a radius that has the same length as AM.
Notice that AC is a diameter of the circle. Now we will draw BD, which will be a diameter of the circle as well. Remember that it will not be on the perpendicular bisector of AC.
AC and BD are congruent and bisect each other. because both of them are diameters of the circle. Lastly, by using a straight edge we will draw a quadrilateral whose edges are A, B, C, and D.
b We will construct a quadrilateral ABCD which has the below properties.
AC and BD bisect each other.
AC and BD are not congruent.
AC and BD are perpendicular.
First, we need to classify the quadrilateral according to the given properties. To do so we will draw a table containing all special quadrilaterals and we will consider whether these quadrilaterals satisfy the properties.
A rhombus meets all of the properties. Therefore, ABCD is a rhombus. Now, let's construct a rhombus. We can begin by drawing the segment AC with the help of a straight edge.
We want to draw BD, which bisects AC and is perpendicular to AC. From here we will find the perpendicular bisector of AC using a compass. Then, we will draw BD which is on the perpendicular bisector. Let's put the compass on point A and draw an arc that passes through point C.
We will use the same compass settings and draw another arc by putting the compass on point C. After that, we will label the intersection points of the arcs as E and F.
Now, we will draw EF which is the perpendicular bisector of AC. We know that EF bisects AC at its midpoint, M.
From here, we will determine BD. To do so, we will draw a circle whose center is at point M and whose radius is not equal to AM. We will label the intersection points of the circle and EF as B and D.
Notice that AC and BD are perpendicular bisectors of each other. Lastly, by using a straight edge we will draw a quadrilateral whose vertices are A, B, C, and D.
