Question

3. Find the circumcenter of triangle ABC with A(1,6), B(1,4), C(5,4). (1point). a)(5,3). b)(3,5). c)(7.3). d)(1,7). 5.What is the name of the segment inside the large triangle?. a)perpendicular bisector. b)altitude. c)median. d)angle bisector. (I have to draw the picture for number 5)

Answer 1

The answer is the option B

The circumcenter of triangle ABC is the point
`(3,5)`

Explanation:

we know that

The circumcenter of a triangle, is the point where the perpendicular bisectors of a triangle meets

In this problem we have the coordinates of the triangle ABC

`A(1,6) ,B(1,4),C(5,4)`

Step 1

Find the slope of the side AB

The side AB is a vertical side (parallel to the y-axis)

The slope of the side AB is undefined

we know that

The perpendicular line to the side AB will be a horizontal line (parallel to the x-axis)

The equation of the perpendicular bisector to the side AB will be the y-coordinate of the midpoint AB

Step 2

Find the y-coordinate of midpoint AB

we know that

The formula to calculate the y-coordinate of the midpoint between two points is equal to

`y=(y1+y2)/(2)`

we have

`A(1,6) ,B(1,4)`

Substitute the values

`y=(6+4)/(2)=5`

therefore

The equation of the perpendicular bisector to the side AB is

`y=5` ------> equation A

Step 3

Find the slope of the side BC

The side BC is a horizontal side (parallel to the x-axis)

The slope of the side BC is zero

we know that

The perpendicular line to the side BC will be a vertical line (parallel to the y-axis)

The equation of the perpendicular bisector to the side BC will be the x-coordinate of the midpoint BC

Step 4

Find the x-coordinate of midpoint BC

we know that

The formula to calculate the x-coordinate of the midpoint between two points is equal to

`x=(x1+x2)/(2)`

we have

`B(1,4),C(5,4)`

Substitute the values

`x=(1+5)/(2)=3`

therefore

The equation of the perpendicular bisector to the side BC is

`x=3` ------> equation B

Step 5

Find the circumcenter of triangle ABC

To calculate the circumcenter

Solve the system of equations compound by equation A and equation B

`y=5` ------> equation A

`x=3` ------> equation B

The intersection point is
`(3,5)` -------> the circumcenter of triangle ABC

see the attached figure to better understand the problem

Answer 2

For problem number 3, the answer is (3, 5). The correct answer between all the choices given is the second choice or letter B.

For problem number 5, the answer is median. The correct answer between all the choices given is the third choice or letter C.

I am hoping that these answers have satisfied your queries and it will be able to help you, and if you’d like, feel free to ask another question.