Basic Constructions

Midpoint & Distance Formula

Follow along and take notes on the following Midpoint Examples.

After this video, we will try a few examples.

Multiple Choice

Which calculation would determine the coordinates of the midpoint of the segment that connects

$$\left(-2,7\right)$$   and   $$\left(6,-5\right)$$  

• $$\left(\frac{-2+6}{2},\frac{7+-5}{2}\right)$$  

• $$\left(\frac{-2+7}{2},\frac{6+-5}{2}\right)$$  

• $$\left(\frac{-2+-5}{2},\frac{7+6}{2}\right)$$  

• $$\left(-2+6,7+-5\right)$$  

Multiple Choice

Complete the question to determine the coordinates of the midpoint of the segment that connects

$$\left(-2,7\right)$$   and   $$\left(6,-5\right)$$  

• $$\left(2,1\right)$$  

• $$\left(\frac{5}{2},\frac{1}{2}\right)$$  

• $$\left(-\frac{7}{2},\frac{13}{2}\right)$$  

• $$\left(4,2\right)$$  

Follow along and take notes on the following Distance Formula Examples.

After this video, we will try a few examples.

Multiple Choice

Which calculation would determine the length of the segment that connects

$$\left(-2,7\right)$$   and   $$\left(3,-5\right)$$  

• $$d=\sqrt{\left(3--2\right)^2+\left(-5-7\right)^2}$$  

• $$d=\sqrt{\left(3--5\right)^2+\left(-2-7\right)^2}$$  

• $$d=\sqrt{\left(3--7\right)^2+\left(-2-5\right)^2}$$  

Multiple Choice

Complete the question to determine the length of the segment that connects

$$\left(-2,7\right)$$   and   $$\left(3,-5\right)$$  

• $$d=\sqrt{\left(5\right)^2+\left(-12\right)^2}=13$$  

• $$d=\sqrt{\left(8\right)^2+\left(-9\right)^2}=\sqrt{145}$$  

• $$d=\sqrt{\left(10\right)^2+\left(-7\right)^2}=\sqrt{149}$$  

Multiple Choice

Determine the Midpoint and the Length of the graphed Line Segment:

• $$M\left(\frac{3}{2},-\frac{5}{2}\right)$$ and $$d=\sqrt{74}$$

• $$M\left(\frac{7}{2},-\frac{5}{2}\right)$$ and $$d=\sqrt{24}$$

• $$M\left(-2,0\right)$$ and $$d=\sqrt{74}$$

Multiple Choice

Which of the following is the correct midpoint formula given two endpoints (x₁, y₁) and (x₂, y₂)?

Multiple Choice

Given two points (x₁, y₁), (x₂, y₂) the distance between them is:

Multiple Choice

How far is point A from point B?

• 7.1
• 7.8
• 8.7
• 8.9

Multiple Choice

Point M with coordinates (3,4) is the midpoint of the line AB and A has the point (-1,6). What is the point of B?

• (1,5)
• (2,10)
• (7,2)
• (1,2)

Multiple Choice

What is the midpoint between (3, -1) and (8, -6)

• (11, -7)
• (-5.5, -3.5)
• (5.5, -3.5)
• (2, -2)

Multiple Choice

Find the y-coordinate of the midpoint of the line

• 1
• 2
• -2
• 3

Multiple Choice

What is the midpoint between (3, -1) and (7, -5)

• (5, -2)
• (10, -6)
• (5, -3)
• (2, -2)

Multiple Choice

Find the distance between the points (4, 3) and (0, 6).

• 3
• 4
• 5
• 7

Multiple Choice

What is the length of the line segment whose endpoints are (1,−4) and (9,2)?

• 5

• $$2\sqrt{17}$$

• 10

• $$2\sqrt{26}$$

Multiple Choice

The coordinates of point R are (−3,2) and the coordinates of point T are (4,1). What is the length of RT ?

• $$2\sqrt{2}$$

• $$5\sqrt{2}$$

• $$4\sqrt{10}$$

• $$\sqrt{10}$$

How to find the slope between two points

how to find the slope between two points

Multiple Select

Find the slope of the line shown in the graph.

• 0

• No slope.

• Undefined.

• 1

Multiple Choice

Find the slope of the line in the graph.

• 0

• No slope.

• Undefined

• 1

Multiple Choice

Find the slope given the points (-2,-9) and (-1,-3). 

• 4
• -1/6
• -6
• 6

Multiple Choice

Find the slope given the points (2, −7) and (−1, 6). 

• -13/3
• 13/3
• 0
• 2/3

Multiple Choice

Find the slope

• -5/3
• -3/5
• 5/3
• 3/5

Multiple Choice

Find the slope:

• 3/4
• 4/3
• -3/4
• -4/3

Parallel and Perpendicular Lines

Parallel vs. Perpendicular

Multiple Choice

Parallel lines have the ______ slope.

• Different

• Same

• Negative

• Opposite Reciprocal

Multiple Choice

Perpendicular lines have the ______ slope.

• Different

• Same

• Negative

• Opposite Reciprocal

Multiple Choice

Perpendicular lines make a _____ angle.

• Corresponding

• Vertical

• Right

• Adjacent

Multiple Choice

What is the negative reciprocal of 2?

• -2

• $$-\frac{1}{2}$$

• $$\frac{1}{2}$$

• 4

Multiple Choice

What is the slope of a line perpendicular to a line with slope  $$\frac{3}{4}?$$  

• $$-\frac{4}{3}$$  

• $$\frac{4}{3}$$

• $$\frac{3}{4}$$

Multiple Choice

What is the negative reciprocal of slope 0?

• 1

• -1

• Undefined

• 0

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(-2, 3), B(2, 6), C(-1, 0), D(3, 3)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(0, 2), B(5, 4), C(1, 8), D(3, 3)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(-9, -12), B(-2, 2), C(-1, 6), D(-5, -2)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(-1, 8), B(2, 6), C(-1, 2), D(3, 3)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(2, 3), B(-1, 4), C(-5, 3), D(-4, 6)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(-3, 13), B(4, -15), C(-2, 5), D(1, -7)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(9, 2), B(-1, 8), C(-5, 16), D(-8, 11)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(-3, 8), B(3, 2), C(7, 1), D(5, -1)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(5, -8), B(-2, -10), C(-6, -13), D(-2, 1)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Use slope to determine if lines AB and CD are parallel, perpendicular, or neither.

A(-4, 7), B(-2, 6), C(2, -2), D(-8, 3)

• Parallel

• Perpendicular

• Neither

Multiple Choice

Which of the following is written in slope-intercept form?

• $$2y+3x=5$$

• $$y=2+x+1$$

• $$y=\frac{1}{2}x-3$$

Multiple Choice

What is the slope of

$$y=3x+5$$  ?

• 3

• y

• 5

• x

Multiple Choice

What is the y-intercept of

$$y=3x+5$$  

• 3

• x

• 5

• y

Perpendicular Lines

• Have opposite reciprocal slopes

• Can have the same y-intercept

• Intersect at a 90 degree angle

Opposite Reciprocal

• Flip the numbers

• Flip the sign

Multiple Choice

The slopes of perpendicular lines are...

• Same

• Opposite Reciprocals

• Opposite

• Reciprocals

Multiple Choice

The opposite reciprocal to $$\frac{5}{2}$$  is?

• $$\frac{2}{5}$$  

• $$-\frac{5}{2}$$  

• $$\frac{5}{2}$$  

• $$-\frac{2}{5}$$  

Multiple Choice

Which slope would be perpendicular to a slope of $$2$$  ?

• $$2$$  

• $$\frac{1}{2}$$  

• $$-\frac{1}{2}$$  

• $$-2$$  

Multiple Choice

Which equation is perpendicular to  $$y=\frac{2}{3}x-4$$ ?

• $$y=-\frac{3}{2}x+4$$  

• $$y=-\frac{2}{3}x+4$$  

• $$y=\frac{2}{3}x-4$$  

• $$y=\frac{3}{2}x-4$$  

Multiple Choice

Are the given lines parallel, perpendicular, or neither?

$$y=\frac{1}{2}x+2$$  and $$y=\frac{1}{2}x-4$$  

• Parallel

• Perpendicular

• Neither

Multiple Choice

Are the given lines parallel, perpendicular, or neither?

$$y=2x+3$$  and  $$y=-\frac{1}{2}x+3$$  

• Parallel

• Perpendicular

• Neither

Equations of Circles

From the Equation

• Make the signs on h and k OPPOSITE to get the center

• Take the SQUARE ROOT of r² to get the radius

• If you don't SEE a h or k, then that means the coordinate is ZERO.

Multiple Choice

In the equation (x-3)²+(y-2)²=16, the center of the circle is...

• (3,2)

• (-3, -2)

• (-2, -3)

• (2, 3)

Multiple Choice

In the equation (x-3)²+(y-2)²=16, the radius of the circle is...

• 16

• 32

• 8

• 4

Multiple Choice

In the equation x²+(y+5)²=10, what is the center?

• (1, 5)

• (-1, -5)

• (0, 5)

• (0, -5)

To Write the Equation

• Make the signs on the center OPPOSITE before you plug them in for h and k.

• SQUARE the radius (multiply it by itself) before you plug it in for r²

• If you were given the diameter, then divide that by 2 to get the radius. And then square it before you plug it in as r²

Multiple Choice

The diameter of a circle has length 12. The center is at (-5, 2). Give the equation of the circle.

• (x - 2)² + (y + 5)² = 36

• (x - 5)² + (y + 2)² = 6

• (x + 5)² + (y - 2)² = 36

• (x + 2)² + (y - 5)² = 6

Multiple Choice

Write the equation of a circle with center (7, 0) with radius 3.

• (x - 7)² + y² = 9

• x² + (y -7)² = 9

• (x - 7)² + y² = 3

• x² + (y -7)² = 3

From a Graph

• Find the coordinates of the center. Make the signs on both of them OPPOSITE before you plug them in for h and k.

• Find the radius by counting the distance from the center out to the edge of the circle. Then SQUARE that before you plug it in for r²

• 
  

Multiple Choice

What is the equation for this circle?

• (x+1)²+(y+1)²=9

• x²+y²=9

• x²+y²=6

• x²+y²=3

Multiple Choice

What is the equation of the circle?

• (x+1)² + (y +1)² = 3

• (x+1)² + (y -1)² = 3

• (x+1)² + (y +1)² = 9

• (x-1)² + (y -1)² = 9

Constructions in Geometry

by Brandy Stapleton

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Multiple Choice

Which one is the correct construction for a perpendicular bisector?

• 1

• 2

• 3

• 4

Multiple Choice

What does the word BISECT mean?

• To cut something into more than five pieces.
• It is a plane with two sets of wings.
• A shape that has three sides. 
• To cut something into two congruent pieces or in half.

Multiple Choice

What must be true?

• 1

• 2

• 3

• 4

Multiple Choice

Use the picture

• 1
• 2
• 3
• 4

Multiple Choice

Select the best answer choice.

• A
• B
• C
• D

Multiple Choice

Which of the following constructions is illustrated?

• An angle is congruent to a given angle
• The bisector of a given angle
• The bisector of a given segment
• The perpendicular bisector of a given segment.

Multiple Choice

To inscribe a square inside a circle, first you must draw a diameter anywhere across the circle. What should your next step be?

• Construct a perpendicular bisector
• Draw a second diameter to the circle
• Construct a line tangent to the circle
• Set your compass the length of the radius.

Multiple Choice

The picture represents a compass and straightedge construction of _________?

• Copy an angle
• Bisect an angle
• Copy a line segment
• Bisect a line segment

Multiple Choice

To inscribe a hexagon inside a circle, which length should you set your compass to?

• The radius
• The diameter
• Half of the radius
• It depends on the size of the circle

Multiple Choice

Nate is constructing angle bisector to angle ABC.  Where did he position his compass to create the two arcs that intersect at the interior of the angle?

• He fixed his compass on B and drew both arcs.
• He fixed his compass on points A and C.
• He fixed his compass on points D and E.
• He fixed his compass on points B and E.

Multiple Choice

Eric was attempting to construct a perpendicular bisector to the segment AB with a compass and straight edge. Which of the below statements explains what Eric may have done wrong?

• Eric should have started by putting the compass needle point at the midpoint of the segment AB.

• On the second step, Eric should have placed the compass needle point where the first arc intersected the segment AB.

• Erick just needed to open the compass more to create arcs that have a radius of more than half the length of the segment AB.

• Erick didn’t do anything wrong he just needs to connect the opposite endpoints of each arc to finish the construction.

Multiple Choice

When constructing a line parallel to a given line, you will be

• copying a segment
• bisecting a segment
• copying an angle
• constructing a perpendicular 

Multiple Choice

Which is the basis for the procedure used in the construction of a line parallel to a given line through a point not on the line?

• Corresponding Angles Theorem

• Alternate Exterior Angles Theorem

• Alternate Interior Angles Theorem

• Consecutive Interior Angles Theorem

Multiple Choice

Suppose we wish to construct angle EFG congruent to angle DBC using a compass and straightedge. Which step would be correct to do first?

• Place the compass point at B
• Place the compass point at C
• Place the straightedge along A and C
• Place the straightedge along C and D

Multiple Choice

What kind of construction is displayed in this picture?

• A circle inscribed in a hexagon
• A hexagon inscribed in a circle
• A hexagon
• A circle