Question

1. what type of triangle center is point a? 10. which point of concurrency is shown below? 11. point m is a centroid. solve for x. (3x + 18) (5x - 22) 12. sketch △jkl with centroid r. make sure to shown all congruency and angle markings.

Explanation:

Step1: Recall centroid property

The centroid divides each median in a 2:1 ratio. If \(M\) is the centroid, the two - segment lengths on the median are related such that \(3x + 18=\frac{1}{2}(5x - 22)\) (since the shorter segment from the centroid to the mid - point of the side is half of the longer segment from the vertex to the centroid). Multiply both sides by 2 to get \(2(3x + 18)=5x - 22\).
\[2(3x + 18)=5x - 22\]
\[6x+36 = 5x - 22\]

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(6x - 5x+36=5x - 5x - 22\), which simplifies to \(x+36=-22\). Then subtract 36 from both sides: \(x=-22 - 36\).
\[x=-58\]

1. Since the angles at the vertex are marked as equal, point \(A\) is the in - center (the point of concurrency of the angle bisectors of a triangle).

1. The point shown in the circle - triangle figure is the circum - center (the point of concurrency of the perpendicular bisectors of the sides of a triangle, which is equidistant from the vertices of the triangle and is the center of the circum - circle).

1. To sketch \(\triangle{JKL}\) with centroid \(R\):

• First, draw a triangle \(\triangle{JKL}\).
• Recall that the centroid is the point of concurrency of the medians. A median is a line segment that joins a vertex of a triangle to the mid - point of the opposite side.
• Mark the mid - points of the sides of the triangle (\(M_1\), \(M_2\), \(M_3\)).
• Draw the medians from each vertex (\(J\), \(K\), \(L\)) to the mid - points of the opposite sides. The point of intersection of these medians is the centroid \(R\). Mark congruency and angle markings as required (e.g., if there are equal sides or equal angles given in the problem, mark them accordingly).

Answer:

1. \(x=-58\)
2. In - center
3. Circum - center
4. Sketch as described above with centroid \(R\) and appropriate markings.