Lesson 7

Distances and Parabolas

View Student Lesson

Problem 1

The point \((6,y)\) is the same distance from \((4,1)\) as it is from the \(x\)-axis. What is the value of \(y\)?

Solution

For access, consult one of our IM Certified Partners.

Problem 2

A parabola is defined as the set of points the same distance from \((6,2)\) and the line \(y=4\). Select all points that are on this parabola.

A:

\((1,\text-2)\)

B:

\((2,\text-1)\)

C:

\((6,2)\)

D:

\((7,3)\)

E:

\((8,2)\)

Solution

For access, consult one of our IM Certified Partners.

Problem 3

Compare and contrast the parabolas with these definitions.

  • parabola A: points that are the same distance from \((0,4)\) and the \(x\)-axis
  • parabola B: points that are the same distance from \((0,\text-6)\) and the \(x\)-axis

Solution

For access, consult one of our IM Certified Partners.

Problem 4

Find the center and radius of the circle represented by the equation \(x^2+y^2-8y+5=0\).

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 6.)

Problem 5

Match each expression with the value needed in the box in order for the expression to be a perfect square trinomial.

A:

​​​​\(x^2 + 14x + \boxed{\phantom{3}}\)

B:

​​​​\(x^2 -\frac12 x + \boxed{\phantom{3}}\)

C:

\(c^2 - 10c + \boxed{\phantom{3}}\)

D:

​​​​\(z^2 +z + \boxed{\phantom{3}}\)

1:

\(\frac{1}{16}\)

2:

49

3:

25

4:

\(\frac14\)

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 6.)

Problem 6

Write each expression as the square of a binomial.

  1. \(x^2-12x+36\)
  2. \(y^2+8y+16\)
  3. \(w^2-16w+64\)

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 5.)

Problem 7

Write an equation of a circle that is centered at \((1,\text-4)\) with a radius of 10.

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 4.)

Problem 8

The density of water is 1 gram per cm3. An object floats in water if its density is less than water’s density, and it sinks if its density is greater than water’s. Will a solid bar of soap shaped like a rectangular prism with mass 1.048 kilograms and dimensions 5.6 centimeters, 13 centimeters, and 16 centimeters float or sink? Explain your reasoning.

Solution

For access, consult one of our IM Certified Partners.

(From Unit 5, Lesson 17.)

Problem 9

Jada has this idea for bisecting angle \(ABC\). First she draws a circle with center \(B\) through \(A\). Then she constructs the perpendicular bisector of \(AD\).

Circle with center point B, central angle ABD, DA segment, point C is on BD outside of the circle.
Circle with center point B, central angle ABD, DA segment, diameter drawn through B and the midpoint of DA, point C is on BD outside of the circle.

Does Jada's construction work? Explain your reasoning. You may assume that the perpendicular bisector of line segment \(AD\) is the set of points equidistant from \(A\) and \(D\).

Solution

For access, consult one of our IM Certified Partners.

(From Unit 1, Lesson 5.)