Which points are on the perpendicular bisector of the given segment? check all that apply.

Video Transcript

which points are on the perpendicular by sector of the given segment. So first we need to find the midpoint, So we have negative 20 and five and 10. 5. 10, 15. It's our main point is negative 10 plus 10 divided by two And five plus 15 divided by two. So this comes out to be -10 divided by two 20, divided by two. So our midpoint is negative five and 10 -5 and 10. So our midpoint is here and the perpendicular by sector, it's going to have a negative reciprocal slopes. So we need to find the slope of these two points. So our slope is 15 -5 Over 10 -120. So This becomes 10/30 Which is 1/3 are perpendicular slope is going to be -3. Mhm. It's going to be -3. So going by tens. So it's gonna go down 5, 10, 15, 20 30, that's 5 2. So each of these are growing by five, It's going down by three. We could change this to negative 30/10. 5. 10. 15, 25, 10, 15, 25. That would be another point on our perpendicular by sector. So are perpendicular by sector. So we're gonna find the equation of the perpendicular by sector. So we have y is equal to negative three x plus b. And it has to pass through this point. So this is 10 is equal to negative three times negative five plus B. So we have 10 is equal to 15 plus B. So we have negative five is equal to be, sorry, equation is y. Is equal to negative three X minus five. So now we're gonna check to see which of these points is on that line. So we have mhm. We have um -3 times negative 8 -5. So negative three Times -8 -5. That's 19. So that is on the perpendicular by sector And then we have negative three times 1 -5. So negative three times 1 -5. That's negative aid. So that's also on the perpendicular by sector. If we plug in zero, that's not on the perpendicular by sector, -3 times negative 5 -5. -3 times negative 5 -5. That's 10. So that's on the perpendicular by sector and then negative three times two minus five. So negative three times two minus Hi, That's not 11. So that's not in the perpendicular by sector.

A geometry class is asked to find the equation of a line that is parallel to y - 3 = −(x + 1) and passes through (4, 2). Trish states that the parallel line is y - 2 = -1(x - 4). Demetri states that the parallel line is y = -x + 6.
Are the students correct? Explain.

Trish is the only student who is correct; the slope should be -1, and the line passes through (4, 2).

Demetri is the only student who is correct; the slope should be -1, and the y-intercept is 6.

Both students are correct; the slope should be -1, passing through (4, 2) with a y-intercept of 6.

Neither student is correct; the slope of the parallel line should be 1.

Which points are on the perpendicular bisector of the given segment? Check all that apply. -8,19 1,-8 0,19 -5,10 2,-7

Question

Gauthmathier6789

Grade 12 · 2022-09-20

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Which points are on the perpendicular bisector of the given segment? Check all that apply.
Which points are on the perpendicular bisector of - Gauthmath
(1,-8)
(0,19)
(-5,10)
(2,-7)

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When a line divides another line segment into two equal halves through its midpoint at 90º, it is called theperpendicularof that line segment. The perpendicular bisector theorem states that any point on the perpendicular bisector is equidistant from both the endpoints of the line segment on which it is drawn. If a pillar is standing at the center of a bridge at an angle, all the points on the pillar will be equidistant from the end points of the bridge.

What is Perpendicular Bisector Theorem?

The perpendicular bisector theorem states that any point on the perpendicular bisector is equidistant from both the endpoints of the line segment on which it is drawn.

In the above figure,

MT = NT

MS = NS

MR = NR

MQ = NQ

What is the Converse of Perpendicular Bisector Theorem?

The converse of the perpendicular bisector theorem states that if a point is equidistant from both the endpoints of the line segment in the same plane, then that point is on the perpendicular bisector of the line segment.

In the above image,  XZ=YZ 

It implies ZO is the perpendicular bisector of the line segment XY.

Proof of Perpendicular Bisector Theorem

Let us look at the proof of the above two theorems on a perpendicular bisector.

Perpendicular Bisector Theorem Proof

Consider the following figure, in which C is an arbitrary point on the perpendicular bisector of AB (which intersects AB at D):

Compare \(\Delta ACD\) and \(\Delta BCD\). We have:

1. AD = BD
2. CD = CD (common)
3. ∠ADC =∠BDC = 90°

We see that \(\Delta ACD \cong \Delta BCD\) by the SAS congruence criterion. CA = CB,which means that C is equidistant from A and B.

Note: Refer to the SAS congruence criterion to understand why \(\Delta ACD\) and \(\Delta BCD\) are congruent.

Perpendicular Bisector Theorem Converse Proof

Consider CA = CB in the above figure.

To prove that AD = BD.

Draw a perpendicular line from point C that intersects line segment AB at point D.

Now, compare \(\Delta ACD\) and \(\Delta BCD\). We have:

1. AC= BC
2. CD = CD(common)
3. ∠ADC = ∠BDC = 90°

We see that \(\Delta ACD \cong \Delta BCD\) by the SAS congruence criterion. Thus, AD = BD, which means that C is equidistant from A and B.

 Important Notes

• The perpendicular bisector theorem and its converse can be proved by the SAS congruency criterion.
• The perpendicular bisector theorem is used in the construction of buildings, bridges, etc., and in making designs where we need to build something in the center and at equal distance from the endpoints.

 Related Topics on Perpendicular Bisector Theorem

• Angle Bisector
• Constructing an angle of 90 degrees
• Congruent Triangles
• How do You Know if Two Line Segments are Perpendicular?

1. Example 1: In a pyramid, line segment AD is the perpendicular bisector of triangle ABC on line segment BC. If AB = 20 feet and BD= 7 feet, find the length of side AC.

   

   Solution

   It is given that AD is the perpendicular bisector on the line segment BC.

   So, By Perpendicular Bisector Theorem, any point on line segment AD is at an equal distance from points B and C. It implies, AB = AC

    AC = 20 feet.

2. Example 2: In any equilateral or isosceles triangle, can we say that the vertex between equal sides lies on the perpendicular bisector of the base?

   
   
   Solution

   Draw a perpendicular from vertex X that intersects segment YZ at point \(O\).

   If XY =XZ,

   Then, by converse of perpendicular bisector theorem, it is proved that OY=OZ .

   \(\therefore\) Vertex X lies on the perpendicular bisector of the base YZ of triangle.

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Frequently Asked Questions(FAQs)

What is the Perpendicular Bisector Theorem?

The perpendicular bisector theorem states that any point on the perpendicular bisector is equidistant from both the endpoints of the line segment on which it is drawn.

What is the Angle Bisector Theorem?

The angle bisector theorem states that in a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle.

What is an Example of a Perpendicular Bisector?

The median of a triangle is the line that joins the vertex of the triangle to the midpoint of the opposite side of the vertex. The median of an equilateral triangle is an example of a perpendicular bisector.

What is the Linear Pair Perpendicular Theorem?

The linear pair perpendicular theorem states that if two straight lines intersect at a point and the linear pair of angles they form have an equal measure, then the two lines are perpendicular to each other.

What is the Median of a Triangle?

The median of a triangle is a line segment which joins a vertex to the midpoint of the opposite side, thus bisecting that particular side. Every triangle has three medians which start from each vertex and intersect each other at the centroid of the triangle.

Which of the following describes a perpendicular bisector of a segment?

How do you find the perpendicular bisector of a point?

Which point lies on the perpendicular bisector of the segment with endpoints?

What is the perpendicular bisector of two points?