Weighted Averages: IM Geo.6.15
[url=https://im.kendallhunt.com/HS/teachers/2/6/15/preparation.html]“Weighted Averages”[/url] from IM Geometry © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Geo.6.15 Lesson
- IM Geo.6.15 Lesson: Weighted Averages
- Geo.6.15 Practice
- IM Geo.6.15 Practice: Weighted Averages
IM Geo.6.15 Lesson: Weighted Averages
For the questions in this activity, use the coordinate grid if it is helpful to you.
What is the midpoint of the segment connecting [math]\left(1,2\right)[/math] and [math]\left(5,2\right)[/math]?[br]
What is the midpoint of the segment connecting [math]\left(5,2\right)[/math] and [math]\left(5,10\right)[/math]?[br]
What is the midpoint of the segment connecting [math]\left(1,2\right)[/math] and [math]\left(5,10\right)[/math]?[br]
Point A has coordinates (2,4). Point B has coordinates (8,1).
Find the point that partitions segment [math]AB[/math] in a [math]2:1[/math] ratio.
Calculate [math]C=\frac{1}{3}A+\frac{2}{3}B[/math].[br]
What do you notice about your answers to the first 2 questions?[br]
For 2 new points [math]K[/math] and [math]L[/math], write an expression for the point that partitions segment [math]KL[/math] in a [math]3:1[/math] ratio.
[size=150]Consider the general quadrilateral [math]QRST[/math] with [math]Q=\left(0,0\right)[/math], [math]R=\left(a,b\right)[/math], [math]S=\left(c,d\right)[/math], and [math]T=\left(e,f\right)[/math].[/size][br][br]Find the midpoints of each side of this quadrilateral.[br]
Show that if these midpoints are connected consecutively, the new quadrilateral formed is a parallelogram.
Here is quadrilateral ABCD.
[list][*]Find the point that partitions segment [math]AB[/math] in a [math]1:4[/math] ratio. Label it [math]B'[/math].[br][/*][*]Find the point that partitions segment [math]AD[/math] in a [math]1:4[/math] ratio. Label it [math]D'[/math].[br][/*][*]Find the point that partitions segment [math]AC[/math] in a [math]1:4[/math] ratio. Label it [math]C'[/math].[br][/*][/list][br]Is [math]AB'C'D'[/math] a dilation of [math]ABCD[/math]? Justify your answer.[br]
IM Geo.6.15 Practice: Weighted Averages
[size=150]Consider the parallelogram with vertices at [math]\left(0,0\right)[/math], [math]\left(4,0\right)[/math], [math]\left(2,3\right)[/math], and [math]\left(6,3\right)[/math]. Where do the diagonals of this parallelogram intersect?[/size]
[size=150]What is the midpoint of the line segment with endpoints [math]\left(1,-2\right)[/math] and [math]\left(9,8\right)[/math]?[/size]
[size=150]Graph the image of triangle [math]ABC[/math] under a dilation with center [math]A[/math] and scale factor [math]\frac{2}{3}[/math].[/size]
[size=150]A quadrilateral has vertices [math]A=\left(0,0\right)[/math], [math]B=\left(2,4\right)[/math], [math]C=\left(0,5\right)[/math], and [math]D=\left(-2,1\right)[/math]. Prove that [math]ABCD[/math] is a rectangle. You may use the applet below to help you.[/size][br]
[size=150]A quadrilateral has vertices [math]A=\left(0,0\right)[/math], [math]B=\left(1,3\right)[/math], [math]C=\left(0,4\right)[/math], and [math]D=\left(-1,1\right)[/math]. [/size][br][br]Select the most precise classification for quadrilateral [math]ABCD[/math].
[size=150]Write an equation whose graph is a line perpendicular to the graph of [math]x=-7[/math] and which passes through the point [math]\left(-7,1\right)[/math].[/size]
Graph the equations (x+1)²+(y-1)²=64 and y=1.
Where do they intersect?
A parabola has a focus of (2,5) and a directrix of y=1.
Decide whether [math](\text{-}2,5)[/math] point on the list is on this parabola. Explain your reasoning.[br]
Decide whether [math](2,3)[/math] point on the list is on this parabola. Explain your reasoning.
Decide whether [math](6,6)[/math] point on the list is on this parabola. Explain your reasoning.