Lesson 5

Squares and Circles

  • Let’s see how the distributive property can relate to equations of circles.

Problem 1

Match each quadratic expression with an equivalent expression in factored form.​​​​​​

A:

\(x^2+6x\)

B:

\(x^2+6x+5\)

C:

\(x^2+6x-7\)

D:

\(x^2+6x+8\)

E:

\(x^2+6x+9\)

1:

\((x+7)(x-1)\)

2:

\((x+5)(x+1)\)

3:

\((x+4)(x+2)\)

4:

\((x+3)(x+3)\)

5:

\(x(x+6)\)

Problem 2

An equation of a circle is \( x^2 - 8x + 16 + y^2 + 10y + 25 = 81\).

  1. What is the radius of the circle?
  2. What is the center of the circle?

Problem 3

Write 3 perfect square trinomials. Then rewrite them as squared binomials.

Problem 4

Write an equation of the circle that has a diameter with endpoints \((12,3)\) and \((\text-18,3)\).

(From Unit 6, Lesson 4.)

Problem 5

  1. Graph the circle \((x-2)^2+(y-1)^2=25\).
  2. For each point, determine if it is on the circle. If not, decide whether it is inside the circle or outside of the circle.
    1. \((4,0)\)
    2. \((\text-3,3)\)
    3. \((\text-2,\text-2)\)
  3. How can you use distance calculations to decide if a point is inside, on, or outside a circle?
Blank coordinate plane with grid, \(x y\) axis, origin \(O\).
(From Unit 6, Lesson 4.)

Problem 6

The triangle whose vertices are \((2,5), (3,1),\) and \((4,2)\) is transformed by the rule \((x,y) \rightarrow (x-2,y+4)\). Is the image similar or congruent to the original figure?

A:

The image is congruent to the original triangle.

B:

The image is similar but not congruent to the original triangle.

C:

The image is neither similar nor congruent to the original triangle.

(From Unit 6, Lesson 3.)

Problem 7

Technology required. A triangular prism has height 6 units. The base of the prism is shown in the image. What is the volume of the prism? Round your answer to the nearest tenth.

A right triangle. An angle is marked 25 degrees. The side opposite the marked angle is labeled 4.
(From Unit 5, Lesson 15.)