Construct triangle XYZ such that XY = 9 cm, YZ = 7.5 cm, and XZ = 10.5 cm. Construct a circle passing through the vertices X, Y, and Z and measure the diameter.
Answer (2)
To construct triangle XYZ with sides equal to 9 cm, 7.5 cm, and 10.5 cm, draw the base XY and use arcs to locate point Z. Construct the circumcircle by finding the circumcenter and measuring its diameter using the circumradius formula. Follow these steps for an accurate construction and understanding of triangle geometry.
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To construct triangle XYZ given the side lengths XY = 9 cm, YZ = 7.5 cm, and XZ = 10.5 cm, and then draw a circle passing through vertices X, Y, and Z - known as the circumcircle - we can follow these steps:
Draw the Triangle:
a. Begin by drawing a straight line segment XY = 9 cm.
b. Set your compass to a radius of 7.5 cm. Place the compass point at Y and draw an arc. This arc represents all the possible positions of point Z at the distance YZ = 7.5 cm.
c. Now, set your compass to a radius of 10.5 cm. Place the compass point at X and draw another arc intersecting the first arc. The intersection of these two arcs is point Z.
d. Connect points X to Z and Y to Z. You now have triangle XYZ.
Construct the Circumcircle (Circle through X, Y, and Z):
a. Find the perpendicular bisector of each side of the triangle.
To find the perpendicular bisector of a segment, place the compass point on one end of the segment, adjust it to slightly more than half the segment's length, and draw arcs above and below the segment. Repeat from the other end of the segment. The intersection of these arcs forms a line that is the perpendicular bisector.
b. The perpendicular bisectors of the sides of a triangle intersect at a single point, the circumcenter. This circumcenter is the center of the circumcircle.
c. Use your compass, set to the distance from this circumcenter to any vertex of the triangle (e.g., X), to draw the circumcircle through all three points X, Y, and Z.
Measure the Diameter of the Circumcircle:
The diameter of the circumcircle is twice the radius. After you have drawn the circumcircle, measure the radius (the distance from the circumcenter to any vertex).
This process will help you construct the desired triangle, draw its circumcircle, and measure the diameter with accuracy.
