A triangle has corners at #(5 , 4 )#, ( 7 , 9)#, and #( 5 , 8 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?

Question

A triangle has corners at #(5 , 4  )#, ( 7 ,   9)#, and #( 5 ,  8 )#.  What are the endpoints and lengths of the triangle's perpendicular bisectors?

Aurora Ayala · Accepted Answer

#" "#
#A(5,4), B(7,9) and C(5,8)#

Bisectors:

#D(6,8.5) and G(7.25,6)# with #DG=2.8 " units"#
#E(6,6.5) and G(7.25,6)# with #EG=1.35 " units"#
#F(5,6) and G(7.25,6)# with #FG=2.25 " units"#

#" "#
Triangle #ABC#:

Vertices: #A(5,4), B(7,9) and C(5,8)#

Construct a perpendicular bisector:

for side #AB#,

find the mid-point: #E(6,6.5)#
#" "#
2. for side #BC#,

find the mid-point: #D(6,8.5)#
#" "#
3. for side #AC#,

find the mid-point: #F(5,6)#
Measure the magnitudes of the perpendicular bisectors:

#EG=1.35" Units"#
#FG=2.25" Units"#
#DG=2.8" units"#
Examine the constructed sketch below and verify:

Hope it helps.

Aaliyah Anderson · Answer

undefined To find the endpoints and lengths of the perpendicular bisectors of the triangle, we need to follow these steps:

1. Find the midpoints of each side of the triangle.
2. Determine the slopes of the sides of the triangle.
3. Find the negative reciprocal of each slope to obtain the slopes of the perpendicular bisectors.
4. Use the midpoint and the slope to find the equation of each perpendicular bisector.
5. Solve for the endpoints of each perpendicular bisector.

Let's go through these steps:

1. Midpoints of each side:
   - Midpoint of (5, 4) and (7, 9): $(6, 6.5)$
   - Midpoint of (7, 9) and (5, 8): $(6, 8.5)$
   - Midpoint of (5, 4) and (5, 8): $(5, 6)$

2. Slopes of each side:
   - Slope of (5, 4) to (7, 9): $m_1 = \frac{9 - 4}{7 - 5} = \frac{5}{2}$
   - Slope of (7, 9) to (5, 8): $m_2 = \frac{8 - 9}{5 - 7} = -\frac{1}{2}$
   - Slope of (5, 4) to (5, 8): $m_3$ is undefined (vertical line)

3. Negative reciprocal of slopes:
   - Negative reciprocal of $m_1$: $m_{\text{perpendicular 1}} = -\frac{1}{m_1} = -\frac{2}{5}$
   - Negative reciprocal of $m_2$: $m_{\text{perpendicular 2}} = -\frac{1}{m_2} = 2$

4. Equation of perpendicular bisectors:
   - For side 1: $y - 6.5 = -\frac{2}{5}(x - 6)$
   - For side 2: $y - 8.5 = 2(x - 6)$
   - For side 3: $x = 5$ (vertical line)

5. Endpoints of perpendicular bisectors:
   - For side 1: Solve the equation with known points (6, 6.5) and (5, 4), yielding the endpoint.
   - For side 2: Solve the equation with known points (6, 8.5) and (7, 9), yielding the endpoint.
   - For side 3: The equation is already a vertical line, so the endpoint can be determined from the given points.

The lengths of these perpendicular bisectors can then be calculated using the distance formula between the endpoints.