# Lesson 15

Weighted Averages

- Let’s split segments using averages and ratios.

### Problem 1

Consider the parallelogram with vertices at \((0,0), (4,0), (2,3),\) and \((6,3)\). Where do the diagonals of this parallelogram intersect?

\((3,1.5)\)

\((4,2)\)

\((2,4)\)

\((3.5,3)\)

### Problem 2

What is the midpoint of the line segment with endpoints \((1,\text-2)\) and \((9,8)\)?

\((3,5)\)

\((4,3)\)

\((5,3)\)

\((5,5)\)

### Problem 3

Graph the image of triangle \(ABC\) under a dilation with center \(A\) and scale factor \(\frac{2}{3}\).

### Problem 4

A quadrilateral has vertices \(A=(0,0), B=(2,4), C=(0,5),\) and \(D=(\text-2,1)\). Prove that \(ABCD\) is a rectangle.

### Problem 5

A quadrilateral has vertices \(A=(0,0), B=(1,3), C= (0,4),\) and \(D=(\text-1,1)\). Select the most precise classification for quadrilateral \(ABCD\).

quadrilateral

parallelogram

rectangle

square

### Problem 6

Write an equation whose graph is a line perpendicular to the graph of \(x=\text-7\) and which passes through the point \((\text-7,1)\).

### Problem 7

Graph the equations \((x+1)^2+(y-1)^2=64\) and \(y = 1\). Where do they intersect?

### Problem 8

A parabola has a focus of \((2, 5)\) and a directrix of \(y=1\). Decide whether each point on the list is on this parabola. Explain your reasoning.

- \((\text{-}1,5)\)
- \((2 ,3)\)
- \((6,6)\)