# Lesson 16

Weighted Averages in a Triangle

### Problem 1

Triangle $$ABC$$ and its medians are shown.

Select all statements that are true.

A:

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

B:

The medians and altitudes are the same for this triangle.

C:

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

D:

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

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

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(From Unit 6, Lesson 15.)

### Problem 5

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

### Solution

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(From Unit 6, Lesson 15.)

### 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

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(From Unit 6, Lesson 14.)

### 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

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(From Unit 6, Lesson 13.)

### 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?

A:

8 units

B:

5 units

C:

3 units

D:

2 units

### Solution

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(From Unit 6, Lesson 8.)