Question

5 gr 3.3 the vertices of △abd are shown. a(-6,7) b(3,5) d(0,3) what are the coordinates of the centroid of △abd? (__,) 6 gr 3.3 the vertices of △tas are shown. t(-1,0) a(-5,-4) s(10,-4) what are the coordinates of the centroid of △tas? (,__) 7 gr 5.3 this question has two parts. obtuse triangle jhi is shown below. part a if the circumcenter of △jhi was constructed, where would it be located? ○ inside of △jhi ○ on the edge of △jhi ○ outside of △jhi ○ you cannot construct a circumcenter for △jhi part b if the incenter of △jhi was constructed, where would it be located? ○ inside of △jhi ○ on the edge of △jhi ○ outside of △jhi ○ you cannot construct an incenter for △jhi

Explanation:

5. Calculate the centroid of \(\triangle ABD\)

Step1: Recall the centroid formula

The centroid formula for a triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) is \((\frac{x_1 + x_2+x_3}{3},\frac{y_1 + y_2 + y_3}{3})\)
Here \(x_1=-6,x_2 = 3,x_3=0\) and \(y_1 = 7,y_2=5,y_3 = 3\)

Step2: Calculate the \(x\) - coordinate of the centroid

\(x=\frac{-6 + 3+0}{3}=\frac{-3}{3}=-1\)

Step3: Calculate the \(y\) - coordinate of the centroid

\(y=\frac{7 + 5+3}{3}=\frac{15}{3}=5\)

6. Calculate the centroid of \(\triangle TAS\)

Step1: Recall the centroid formula

The centroid formula for a triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) is \((\frac{x_1 + x_2+x_3}{3},\frac{y_1 + y_2 + y_3}{3})\)
Here \(x_1=-1,x_2=-5,x_3 = 10\) and \(y_1=0,y_2=-4,y_3=-4\)

Step2: Calculate the \(x\) - coordinate of the centroid

\(x=\frac{-1-5 + 10}{3}=\frac{4}{3}\)

Step3: Calculate the \(y\) - coordinate of the centroid

\(y=\frac{0-4-4}{3}=\frac{-8}{3}\)

7. Properties of circum - center and in - center

Part A: Circum - center of an obtuse triangle

The circum - center of an obtuse triangle lies outside the triangle.

Part B: In - center of a triangle

The in - center of a triangle (acute, obtuse or right - angled) is the intersection of the angle - bisectors of the triangle and it always lies inside the triangle.

Answer:

1. \((-1,5)\)
2. \((\frac{4}{3},-\frac{8}{3})\)
3. Part A: Outside of \(\triangle JHI\); Part B: Inside of \(\triangle JHI\)