1. In triangle ABC, AB=BC=17 and AC=16 Find the circumradius of triangle ABC.
- I have gotten two wrong answers which were both 15/2 and 17/2... Plz halp I do want an explantion just to undersand.
2. In triangle ABC, M is the midpoint of line AB Let D be the point on line BC such that line AD bisects angle BAC, and let the perpendicular bisector of line AB intersect line AD at E If AB=44 and ME=12 then find the distance from E to line AC.
- I have seen similar questions like these posted but I personly cant understand those.
THX in ADVANCE
+23245 1) I'm going to put this on a coordinate axis.
I place point A at the origin: A = (0,0)
I place point C at the point (16,0)
Since this is an isosceles triangle, point B will be somewhere above the midpoint of AC.
The distance from A to this midpoint is 8; the distance from A to point B is 17.
This makes an 8 - 15 - 17 right triangle, so point B = (8,15)
The circumcircle has its center at the intersection points of the perpendicular bisectors of the sides.
The line drawn from point B to the midpoint of AC is one of these perpendicular bisectors and its equation is x = 8.
Now, I plan to find the perpendicular bisector of AB and find where this line crosses the line x = 8.
The slope of AB is: (15 - 0) / (8 - 0) = 15/8.
This means that the perpendicular bisector of AB has a slope of -8/15.
The midpoint of AB is the point (4, 7.5).
The equation of the perpendicular bisector of AB is: y - 7.5 = (-8/15)(x - 4)
Multiplying this out: 15y - 112.5 = -8x + 32 ---> 8x + 15y = 144.5.
Now, to find where this line intersects the line x = 8: 8(8) + 15y = 144.5 ---> 15y = 80.5
---> y = 5.03125
This means that the circumcenter is (8, 5.03125)
To find the radius, I will find the distance from this point to the point (0,0):
distance = sqrt( (8.5 - 0)2 + (5.03125 - 0)2 ) = sqrt( 97.56347656 ) = 9.877422567.
I hope that this is clear and I hope that I didn't mess it up!
+23245 2) If I understand the question correctly:
There is an angle(BAC).
AD bisectes this angle.
E is on AD.
M is a point on AB and ME is perpendicular to AB.
Therefore, the length of ME is the distance from E to line segment AB.
Since every point on the bisector of an angle is equidistant from the sides of the angle, the distance from E to AB
is the same as the distance from E to AC.
Since distances are measured perpendicular to segments, E to AC must equal E to AB, so the distance from E to AC
must also be 12.
1. I understand you reasoning and I thank you for it! I have a hard time understanding decimals tho.. THX I still understand and I will probly use it in the future
2. THX, I thought it was more complicated that that lol!
I might not fully understand 1 but I went from a 0 undersding to a 85 out of 100. Thanks for all of your help!
Acute isosceles triangle.
Sides: a = 17 b = 17 c = 16
Area: T = 120
Perimeter: p = 50
Semiperimeter: s = 25
Angle ∠ A = α = 61.928° = 61°55'39″ = 1.081 rad
Angle ∠ B = β = 61.928° = 61°55'39″ = 1.081 rad
Angle ∠ C = γ = 56.145° = 56°8'42″ = 0.98 rad
Height: ha = 14.118
Height: hb = 14.118
Height: hc = 15
Median: ma = 14.151
Median: mb = 14.151
Median: mc = 15
Inradius: r = 4.8
Circumradius
The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.
R =abc / 4rs
Circumradius: R = 9.633
