The characteristics of the given points are observed during the
geometric construction processes.
Responses:
Step 1: The point O is the center because it is equidistant from the three points
Step 2: PQ = PR = QR, therefore, ΔPQR is an equilateral triangle
b) The ratio of a leg to the hypotenuse side of ΔACF is 0.5, therefore, ΔACF is a 30°– 60°–90° triangle
c) The points of intersection of the diameter and a perpendicular bisector with the circumference of the circle are the vertices an inscribed square of the circle
Which process of geometric construction determines the characteristics of the given points?
Step 1: How to know that the point O is the center of the circle is as follows;
The perpendicular bisectors constructed are the locus of points
equidistant from the both points D and E and also the perpendicular
bisector gives the locus of points equidistant from points E and F
Therefore;
OE = OE, by reflexive property
OE is also equal to OF and OD by definition of the point O being on both
perpendicular bisectors.
Given that OE = OF = OD, we have that the point O is the center of a
circle with radius, OE = OF = OD passing through the points E, F, and D
Therefore;
- The point O is the center because it is equidistant from the three points
Step 2: Given that the center of the circles are the points P and Q, we have;
PQ = The radial length of the two circles
Similarly
PR and QR are radial lengths of circle P and Q respectively
Therefore;
PQ = PR = QR
Which gives;
- The lengths of the sides of ΔPQR are equal and ΔPQR is an equilateral triangle
b) FC is the diameter of circle M
Therefore;
∠ACF is 90° and ΔACF is a right triangle
The ratio of the a leg to the hypotenuse side of a 30°– 60°–90° triangle is 0.5
Length of FA = FM = MC
[tex]\dfrac{FA}{FC} = \dfrac{FA}{FM + FC} = \dfrac{FA}{FA + FA} = \dfrac{FA}{2 \times FA} = \dfrac{1}{2} = \mathbf{0.5}[/tex]
- Given that the ratio of a leg to the hypotenuse side of ΔACF is 0.5, ΔACF is a 30°– 60°–90° triangle
c) The steps that can be used to locate the vertices of a square inscribed
in a circle are;
- Draw a diameter of the circle (passing through the center)
- Construct the perpendicular bisector of the diameter, and extend it to intersect with the circle at the possible two points
- The points of intersection of the diameter and the perpendicular bisector with the circumference of the circle are the location of the vertices an inscribed square of the circle
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