Geometry basics distance and midpoint formulas answer key

Asked by wiki @ 10/06/2021 in Mathematics viewed by 792 People

Name:

Unit 1: Geometry Basics
Date:
Per: Homework 3: Distance & Midpoint Formulas
** This is a 2-page document! **
Directions: Find the distance between each pair of points.
1. 1-4.6) and (3.-7)
2. (-6,-5) and (2.0)
M=(-12,-1)
M=
4. (0.-8) and (3.2)
3. (-1, 4) and (1-1)
5.
.
Directions: Find the coordinates of the midpoint of the segment given its endpoints.
6. /15, 8) and B(-1,-4)
7. M(-5,9) and N[-2.7)
8. P(-3,-7) and Q13.-5)
9. F12.-6) and G(-8,5)
Gina Whion (All Things Algobro. LLC) 2014-2017

Answered by wiki @ 10/06/2021

Answer:

The answer is below

Step-by-step explanation:

The distance between two points is given by the formula:

1) (-4,6) and (3, -7)

2) (-6, -5) and (2,0)

3) (0, -8) and (3, 2)

4) (-1, 4) and (1, -1)

The midpoint (x, y) between two endpoints is given by:

The midpoints of the following segments are:

1) (15, 8) and (-1, -4)

2) M(-5,9) and N[-2.7)

3) P(-3,-7) and Q13.-5)

4)  F(12.-6) and G(-8,5)

Do you know the better answer?

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For any two points in a coordinate plane, the point which lies exactly half way between them can be found using the midpoint formula and the distance between the points can be calculated using the distance formula.

The midpoint of a line segment is the point that divides the segment into two segments of equal length.

The point M is the midpoint of the segment AB since the distance from A to M is the same as the distance from M to B.

A segment bisector is an object that passes through the midpoint of a line segment, bisecting it or dividing it into two segments of equal length. The object can be a point, a segment, a line, a ray, or a plane.

If M is the midpoint of AB, then it is a segment bisector of AB. Furthermore, since line passes through M, it is also a segment bisector of AB.

Identify the segment bisector of AB. Then find the length of AB.

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To begin, notice that is a segment bisector of AB, because it passes through the midpoint, M. ST divides AB into two segments of equal length, AM and MB. Their lengths are given as

We can create an equation by setting the above expressions equal to each other. Then, we can solve for x. When we know x, we can find the total length of AB.

4x−2=2x+1

4x=2x+3

2x=3

x=1.5

We can now find an expression for the length of AB by adding the lengths of AM and MB.

Finally, to find the length of AB we can add AM and MB.

AB=AM+MB=4+4=8 units

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To bisect a segment means to draw the object that bisects it, or passes through its midpoint.

Here, a bisector of AB will be constructed using a compass and a straightedge.

Draw an Arc Centered at A

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Place a compass with its needle point at A. Set the compass to go beyond half the length of the segment. Then draw an arc that intersects the segment.

Draw an Arc Centered at B with the Same Radius

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Keep the compass set to the same length. Then place the needle point at the other end of the segment and draw a second arc across the segment.

If the arcs have been drawn large enough, they will now intersect each other twice. If they don't, extend them.

Draw a Line Through the Points of Intersections of Arcs

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Use the straightedge to draw a line through the intersection points of the arcs. The point where the line and the segment intersect is the midpoint of AB.

Bisectors of AB

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Any line that passes through the midpoint of AB is a bisector of AB.

The midpoint M between two points and on a coordinate plane can be determined by the following formula.

The formula above is called the Midpoint Formula.

For simplicity, the points and will be arbitrarily plotted in Quadrant I. Also, consider the line segment that connects these points. The midpoint M between A and B is the midpoint of this segment. Note that the position of the points in the plane does not affect the proof.

Consider the horizontal distance and the vertical distance between A and B. Since M is the midpoint, M splits each distance, and , in half. Therefore, the horizontal and vertical distances from each endpoint to the midpoint are and Let and be the coordinates of M.

Now, focus on the x-coordinates. The difference between the corresponding x-coordinates gives the horizontal distances between the midpoint and the endpoints.

The graph above shows that these distances are both equal to Therefore, by the Transitive Property of Equality, they are equal.

This equation can be solved to find the x-coordinate of the midpoint M.

The x-coordinate of M is In the same way, it can be shown that the y-coordinate of M is With this information, the coordinates of M can be expressed in terms of the coordinates of A and B.

The segment AB has the endpoints A(-1,6) and B(5,2). Find its midpoint, M.

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We can find the midpoint, M of the segment by using the midpoint formula.

The midpoint of the segment lies at (2,4).

Sometimes, it is necessary to divide a segment into two pieces that are not of equal length. This can be done when the endpoints of the segment and the desired ratio of the lengths are known. For example, find the point P that divides AB so that the ratio of AP to PB is 3:2.

Determine the vertical and horizontal change between A and B.

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To begin, it is necessary to determine the vertical and horizontal change — rise and run — between the points. To do this, count the spaces between A and B in both directions.

Determine the distance from A to P

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Since the desired ratio is 3:2, consider dividing AB into 5 total pieces — AP spans 3 of those pieces, and PB spans 2. Choosing point A as a starting point, P will lie of the way to B.

• If the entire horizontal distance is 8, gives the horizontal distance from A to P — 4.8 units.
• If the entire vertical distance is 7, gives the vertical distance from A to P — 4.2 units.

Now that the distance from A to P is known, P can be added to the graph to so that AB is divided into a ratio of 3:2 at P.

Given two points A(x1,y1) and B(x2,y2) on a coordinate plane, their distance d is given by the following formula.

This formula is called the Distance Formula.

To prove the Distance Formula, and will be plotted on a coordinate plane. For simplicity, both points will be arbitrarily plotted in Quadrant I. Note that the position of the points in the plane does not affect the proof. It will be assumed that x2 is greater than x1 and that y2 is greater than y1.

Next, a right triangle will be drawn. The hypotenuse of this triangle will be the segment that connects points A and B.

The difference between the x-coordinates of the points is the length of one of the legs of the triangle. Furthermore, the length of the other leg is given by the difference between the y-coordinates. Therefore, the lengths of the legs are x2−x1 and y2−y1. Now, consider the Pythagorean Equation.

Here, a and b are the lengths of the legs, and c the hypotenuse of a right triangle. To find the hypotenuse of the right triangle shown in the diagram, the expressions for the legs can be substituted for a and b. Then, the equation can be solved for the hypotenuse c.

Note that, when solving for c, only the principal root was considered. The reason is that c represents the length of a side and therefore must be positive. Keeping in mind that c is the distance between A(x1,y1) and B(x2,y2), then c=d. By the Transitive Property of Equality, the Distance Formula is obtained.

Find the distance between the points A(2,-1) and B(6,2).

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We can find the distance between two points by using the distance formula.

Thus, the distance between the points is 5 units.

To find the perimeter of a shape in the coordinate plane, it is necessary to see each side of the shape as a line segment. When the coordinates of the vertices are known, the length of each side can be calculated separately using the distance formula.

The length of the perimeter is then found by adding the lengths of the sides.

Area is usually calculated using different formulas. However, for some shapes, one formula is not enough. Consider the example shown below.

The shape can be divided into pieces so that the area of each piece can be calculated. Here, that means dividing the shape into a rectangle and a triangle.

Once the vertices are marked, the needed lengths can be found using the distance formula. Then, the area of each shape can be found using the corresponding formula. The total area is the sum of the individual areas.

Find the area and the perimeter of the rectangle.

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To begin, before we can calculate the area and perimeter of the rectangle, we need to know the lengths of the sides. Since the quadrilateral is a rectangle, the top and bottom have equal lengths and the sides have equal lengths.

We'll find CD, the length of the right side, and AD, the length of the bottom. Since the coordiantes are given, we can use the distance formula. We'll determine CD first.

Thus, the length of the sides of the rectangle is units. The length of the top and bottom can be found in the same way.

Let's label the sides with their lengths.

Now that the dimensions of the rectangle are known, the area, A can be found by multiplying the length and the width, and the perimeter, P can be found by adding all four sides.

Thus, the area is 24 square units and the perimeter is approximately 22.6 units.