triangle ABC,angleA=60 degree,angleB=70 degree,angleC=50 degree. AB=6cm,BC=6.5cm,AC=7cm. Find its curcumcentre.
To find the circumcenter of triangle ABC, we need to determine the point where the perpendicular bisectors of the three sides of the triangle intersect. Here's how we can go about finding it:
1. Draw triangle ABC with the given measurements and angles.
2. Locate the midpoints of each side. The midpoint of AB is point M1, the midpoint of BC is point M2, and the midpoint of AC is point M3.
3. Draw the perpendicular bisectors of each side. To do this, draw a line perpendicular to AB passing through point M1. Label the intersection point of this perpendicular bisector with AB as point P1. Repeat this process to find point P2 on the perpendicular bisector of BC and point P3 on the perpendicular bisector of AC.
4. Find the intersection point of the three perpendicular bisectors. This point is the circumcenter of triangle ABC. Label this point as point O.
Now that we have the construction, we need to calculate the coordinates of point O:
5. Let's assume the coordinates of point A are (0, 0).
6. Use the midpoint formula to find the coordinates of the midpoints of each side:
- The coordinates of M1 (midpoint of AB) are ( (x1 + x2)/2 , (y1 + y2)/2 ), where (x1, y1) are the coordinates of A and (x2, y2) are the coordinates of B. Calculate the midpoint of AB accordingly.
- Similarly, find the coordinates of M2 and M3.
7. Determine the slopes of the lines passing through AB, BC, and AC:
- The slope of the line passing through AB is (y2 - y1) / (x2 - x1), where (x1, y1) are the coordinates of A and (x2, y2) are the coordinates of B. Calculate the slope of AB accordingly.
- Similarly, find the slopes of BC and AC.
8. Find the slopes of the perpendicular bisectors:
- The slope of the perpendicular bisector to AB is the negative reciprocal of the slope of AB. Calculate the slope of this perpendicular bisector accordingly.
- Similarly, find the slopes of the perpendicular bisectors to BC and AC.
9. Use the point-slope form to find the equations of the perpendicular bisectors:
- The equation of the line passing through the midpoint M1 (x1, y1) with slope m1 is given by y - y1 = m1(x - x1). Calculate the equations of the perpendicular bisectors accordingly.
- Similarly, find the equations of the perpendicular bisectors for BC and AC.
10. Solve the system of equations formed by the three perpendicular bisectors to find the coordinates of the circumcenter point O. This can be done by solving the equations simultaneously.
Once you have completed these steps, you will have the coordinates of the circumcenter point O, which is the solution to the problem.
To find the circumcenter of triangle ABC, we can follow these steps:
1. Draw triangle ABC with the given measurements.
2. Find the perpendicular bisectors of two sides of the triangle.
3. The point where these two perpendicular bisectors intersect is the circumcenter.
Now, let's calculate the circumcenter of triangle ABC step by step:
Step 1: Draw triangle ABC
Draw a triangle and label the vertices as A, B, and C.
Let AB = 6cm, BC = 6.5cm, and AC = 7cm.
Label the angles as follows:
Angle A = 60 degrees,
Angle B = 70 degrees,
Angle C = 50 degrees.
Step 2: Find the perpendicular bisectors
To find the perpendicular bisectors, draw a line segment perpendicular to each side of the triangle, passing through the midpoint of each side.
Let's label the midpoints:
Midpoint of AB: M,
Midpoint of BC: N,
Midpoint of AC: P.
Step 3: Find the circumcenter
The circumcenter is the point where the perpendicular bisectors intersect.
Let's label the circumcenter as O.
Now, find the equations of the perpendicular bisectors and solve them to find the coordinates of the circumcenter O.
Equation of the perpendicular bisector of AB:
Considering M (midpoint of AB), the slope of AB is (yB-yA)/(xB-xA). In this case, it is (0-6)/(0-6) = -1.
Hence, the slope of the perpendicular bisector is 1/1 = 1.
So, the equation of the perpendicular bisector of AB passing through M is y - yM = m(x - xM), where m is the slope and (xM, yM) are the coordinates of M.
Since M is the midpoint of AB, its coordinates are [(xA+xB)/2, (yA+yB)/2].
In this case, M = [(0+6)/2, (0+0)/2] = [3, 0].
Therefore, the equation of the perpendicular bisector of AB is:
y - 0 = 1(x - 3),
which simplifies to y = x - 3.
Equation of the perpendicular bisector of BC:
Similarly, considering N (midpoint of BC), the slope of BC is (yC-yB)/(xC-xB). In this case, it is (0-6)/(0-6.5) = 12/6.5 = 24/13.
Hence, the slope of the perpendicular bisector is -13/24.
So, the equation of the perpendicular bisector of BC passing through N is y - yN = m(x - xN), where m is the slope and (xN, yN) are the coordinates of N.
Since N is the midpoint of BC, its coordinates are [(xB+xN)/2, (yB+yN)/2].
In this case, N = [(0+6.5)/2, (0+0)/2] = [3.25, 0].
Therefore, the equation of the perpendicular bisector of BC is:
y - 0 = -13/24(x - 3.25),
which simplifies to y = -13x/24 + 13*3.25/24.
Step 4: Find the intersection point
Now, solve the two equations of the perpendicular bisectors to find the coordinates of the circumcenter O.
Substitute the equation of the perpendicular bisector AB into the equation of the perpendicular bisector BC and solve for x:
x - 3 = -13x/24 + 169/24
24x - 72 = -13x + 169
24x + 13x = 169 + 72
37x = 241
x = 241/37 ≈ 6.514
Substitute the value of x into either one of the perpendicular bisector equations to find y:
y = x - 3
y ≈ 6.514 - 3
y ≈ 3.514
Therefore, the coordinates of the circumcenter O are approximately O ≈ (6.514, 3.514).