# Lesson 16

Weighted Averages in a Triangle

### Problem 1

Triangle \(ABC\) and its medians are shown.

Select **all** statements that are true.

The medians intersect at \(\left(\frac{1}{3}, 2\right)\).

The medians and altitudes are the same for this triangle.

An equation for median \(AE\) is \(y=\frac{6}{7}(x+2)\).

Point \(G\) is \(\frac{2}{3}\) of the way from \(A\) to \(E\).

Median \(BF\) is congruent to median \(CD\).

### Solution

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### Problem 2

Triangle \(ABC\) has vertices at \((\text-2,0), (\text-1,6),\) and \((6,0)\). What is the point of intersection of the triangle’s medians?

### Solution

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### Problem 3

Triangle \(EFG\) and its medians are shown.

Match each pair of segments with the ratios of their lengths.

### Solution

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### Problem 4

Given \(A=(\text-3,2)\) and \(B=(7,\text-10)\), find the point that partitions segment \(AB\) in a \(1:4\) ratio.

### Solution

### Problem 5

Graph the image of quadrilateral \(ABCD\) under a dilation using center \(A\) and scale factor \(\frac{1}{3}\).

### Solution

### Problem 6

A trapezoid is a quadrilateral with at least one pair of parallel sides. Show that the quadrilateral formed by the vertices \((0,0), (5,2), (10,10),\) and \((0,6)\) is a trapezoid.

### Solution

### Problem 7

Here are the graphs of the circle centered at \((0,0)\) with radius 6 units and the line given by \(2x+y=11\). Determine whether the circle and the line intersect at the point \((3,5)\). Explain or show your reasoning.

### Solution

### Problem 8

A parabola has focus \((\text-3,2)\) and directrix \(y=\text-3\). The point \((a,5)\) is on the parabola. How far is this point from the focus?

8 units

5 units

3 units

2 units