Lesson 16

Weighted Averages in a Triangle

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Problem 1

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

Triangle ABC graphed. A = -2 comma 0, B = 0 comma 4, C = 3 comma 2. median CD, D = -1 comma 2. median BF, F = 0 point 5 comma 1. median AE, E = 1 point 5 comma 3. 

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.

Triangle EFG and its medians EH, FI, and GJ drawn.

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

A:

\(GK:KJ\)

B:

\(GH:HF\)

C:

\(HK:KE\)

1:

\(1:1\)

2:

\(1:2\)

3:

\(2:1\)

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 \(\frac{1}{3}\).

Quadrilateral ABCD. Point A at 0 comma 0. Point B at 10 comma 6. Point C at 12 comma 18. Point D at negative 6 comma 6.

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.

Graph of circle with center 0 comma 0, radius 6 units. Line is 2x plus y equals 11.

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.)