Question

directions: show any work. 1. draw lines to match the special segment with its corresponding triangle center. median, perpendicular bisector, angle bisector, altitude, incenter, orthocenter, centroid, circumcenter 2. which point of concurrency is the center of an inscribed circle of a triangle like the one shown below? 3. triangle def is an obtuse triangle. for each triangle center, write whether the point would be inside, on, or outside the triangle. incenter - inside, orthocenter - outside, centroid - inside, circumcenter - outside 4. which triangle center is shown below? 5. sketch triangle ghi with orthocenter p. make sure to shown all congruency and angle markings. 6. point g is a circumcenter. solve for x. (2x + 15)°

Explanation:

Step1: Recall triangle - center definitions

Median: A line segment joining a vertex to the mid - point of the opposite side. The centroid is the point of concurrency of the medians.
Perpendicular bisector: A line that is perpendicular to a side and bisects it. The circumcenter is the point of concurrency of the perpendicular bisectors.
Angle bisector: A line that divides an angle of the triangle into two equal angles. The incenter is the point of concurrency of the angle bisectors.
Altitude: A line segment from a vertex perpendicular to the opposite side. The orthocenter is the point of concurrency of the altitudes.
So, median - centroid; perpendicular bisector - circumcenter; angle bisector - incenter; altitude - orthocenter.

Step2: Identify incenter for inscribed circle

The incenter is the center of the inscribed circle of a triangle. It is the point where the angle bisectors of the triangle meet.

Step3: Locate triangle centers in obtuse triangle

In an obtuse - angled triangle:
The incenter (point of concurrency of angle bisectors) is always inside the triangle.
The orthocenter (point of concurrency of altitudes) is outside the triangle.
The centroid (point of concurrency of medians) is inside the triangle.
The circumcenter (point of concurrency of perpendicular bisectors) is outside the triangle.

Step4: Identify triangle center from figure

The figure shows the perpendicular bisectors of the sides of the triangle intersecting at a point. So, the center shown is the circumcenter.

Step5: Sketch triangle with orthocenter

To sketch triangle GHI with orthocenter P:

1. Draw an obtuse - angled triangle (since the orthocenter of an obtuse - angled triangle is outside the triangle).
2. Draw the altitudes from each vertex. The altitudes are lines perpendicular to the opposite sides. The point where these altitudes intersect outside the triangle is labeled as P. Mark congruency and angle markings as required. For example, if two sides are equal, mark them with the same number of hash marks, and if two angles are equal, mark them with the same number of arcs.

Step6: Solve for x using circumcenter property

Since G is the circumcenter, the perpendicular bisectors of the sides of the triangle meet at G. Let's assume that the angle formed by the perpendicular bisector and the side of the triangle is 90 degrees. If we assume that the given angle \((2x + 15)^{\circ}=90^{\circ}\) (since the perpendicular bisector makes a right - angle with the side of the triangle).
Set up the equation \(2x+15 = 90\).
Subtract 15 from both sides: \(2x=90 - 15\).
\(2x=75\).
Divide both sides by 2: \(x=\frac{75}{2}=37.5\).

Answer:

1. median - centroid; perpendicular bisector - circumcenter; angle bisector - incenter; altitude - orthocenter
2. Incenter
3. Incenter - inside; orthocenter - outside; centroid - inside; circumcenter - outside
4. Circumcenter
5. Sketch an obtuse - angled triangle with altitudes intersecting at a point (orthocenter P) outside the triangle and mark congruency and angle markings.
6. \(x = 37.5\)