Completing the Square: IM Geo.6.6
[url=https://im.kendallhunt.com/HS/teachers/2/6/6/preparation.html]“Completing the Square”[/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.6.6 Lesson
- IM Geo.6.6 Lesson: Completing the Square
- Geo.6.6 Practice
- IM Geo.6.6 Practice: Completing the Square
IM Geo.6.6 Lesson: Completing the Square
For each expression, what value would need to be in the box in order for the expression to be a perfect square trinomial?
[math]x^2+10x+ \boxed{ }[/math]
[math]x^2-16x+\boxed{ }[/math] [br]
[math]x^2+40x+\boxed{ }[/math][br]
[math]x^2+5x+\boxed{ }[/math]
[size=150]Here is the equation of a circle: [math]x^2+y^2-6x-20y+105=0[/math][br][br]Elena wants to find the center and radius of the circle. Here is what she’s done so far.[br][br]Step 1: [math]x^2-6x+y^2-20y=-105[/math][br]Step 2: [math]x^2-6x+9+y^2-20y+100=-105+9+100[/math][br]Step 3: [math]x^2-6x+8+y^2-20y+100=4[/math][/size][br][br]What did Elena do in the first step?[br]
Why did Elena add 9 and 100 to the [i]left [/i]side of the equation in Step 2?[br]
Why did Elena add 9 and 100 to the [i]right [/i]side of the equation in Step 2?[br]
What should Elena do next?[br]
What are the center and radius of this circle?[br]
Draw a graph of the circle.
[size=150]Here is the equation of a circle: [math]x^2+y^2-2x+4y-4=0[/math][br][/size][br]Find the center and radius of the circle. Explain or show your reasoning.[br]
Draw a graph of the circle.
Triangulation is a process of using 3 distances from known landmarks to locate an exact position. Find a point that is located 5 units from the point [math]\left(3,4\right)[/math], 13 units from the point [math]\left(11,-4\right)[/math], and 17 units from the point [math]\left(21,16\right)[/math]. Use the coordinate grid if it’s helpful.
IM Geo.6.6 Practice: Completing the Square
Suppose a classmate missed the lessons on completing the square to find the center and radius of a circle. Explain the process to them. If it helps, use a problem you’ve already done as an example.
Match each expression with the value needed in the box in order for the expression to be a perfect square trinomial.
[math]x^2 -8x+\boxed{ }[/math]
[math]x^2 +20x+\boxed{ }[/math]
[math]x^2 -16x+\boxed{ }[/math]
[math]x^2 +9x+\boxed{ }[/math]
[size=150]Find the center and radius of the circle represented by the equation [math]x^2+y^2+4x-10y+20=0[/math][/size].
[size=150] Select[b] all[/b] the expressions that can be factored into a squared binomial.[/size]
[size=150]An equation of a circle is given by [math]\left(x+3\right)^2+\left(y-9\right)^2=5^2[/math]. Apply the distributive property to the squared binomials and rearrange the equation so that one side is 0.[/size]
Graph the circle (x+1)²+(y-3)²=16.
Find the distance from the center of the circle to each point on the list.
[math]\left(2,1\right)[/math]
[math]\left(4,1\right)[/math]
[math]\left(3,3\right)[/math]
What do these distances tell you about whether each point is inside, on, or outside the circle?[br]
[size=150]The triangle whose vertices are [math]\left(3,-1\right),\left(2,4\right)[/math] and [math]\left(5,1\right)[/math] is transformed by the rule [math]\left(x,y\right)\rightarrow\left(2x,5y\right)[/math]. Is the image similar or congruent to the original figure?[/size]
A cube has side length 3 inches. A sphere has a radius of 3 inches.
Before doing any calculations, predict which solid has greater surface area to volume ratio.[br]
Calculate the surface area, volume, and surface area to volume ratio for each solid.[br]