find orthocenter
triangle PQR .P(1,7)Q (1,2)R (11,2).
Two triangles have the following pairs of congruent sides BD=Fj=DG=JM,and GB=MF. Write the congruence statement for the two triangles.
Note that the triangle described is a right triangle. The orthocenter is at the right angle, since the triangle can be inscribed in a circle, with the hypotenuse as the diameter.
Confusion
To find the orthocenter of a triangle, you need to follow these steps:
1. Calculate the slopes of the three sides of the triangle.
2. Find the slopes of the perpendicular bisectors of each side.
3. Determine the point of intersection of the perpendicular bisectors, which will be the orthocenter.
Let's calculate the slopes of the sides first:
Side PQ:
- The slope of a line passing through points P(x1, y1) and Q(x2, y2) is given by the formula: (y2 - y1) / (x2 - x1).
- For points P(1, 7) and Q(1, 2), the slope of side PQ is (2 - 7) / (1 - 1), which is undefined since the denominator is zero.
Side QR:
- For points Q(1, 2) and R(11, 2), the slope of side QR is (2 - 2) / (11 - 1) = 0.
Side RP:
- For points R(11, 2) and P(1, 7), the slope of side RP is (7 - 2) / (1 - 11) = -5/10 = -1/2.
Now, let's find the slopes of the perpendicular bisectors:
Perpendicular bisector of PQ:
- The midpoint of PQ is the average of the coordinates of P and Q: ((1 + 1) / 2, (7 + 2) / 2) = (1, 9/2).
- The negative reciprocal of the slope of side PQ is the slope of the perpendicular bisector of PQ, which is undefined in this case.
Perpendicular bisector of QR:
- The midpoint of QR is the average of the coordinates of Q and R: ((1 + 11) / 2, (2 + 2) / 2) = (6, 2).
- The negative reciprocal of the slope of side QR (which is 0) is undefined, so the perpendicular bisector is a vertical line passing through the midpoint (6, 2).
Perpendicular bisector of RP:
- The midpoint of RP is the average of the coordinates of R and P: ((11 + 1) / 2, (2 + 7) / 2) = (6, 9/2).
- The negative reciprocal of the slope of side RP (-1/2) is 2, so the slope of the perpendicular bisector of RP is 2.
Finally, we need to find the point of intersection of the perpendicular bisectors, which is the orthocenter:
Since the perpendicular bisector of PQ is undefined and the perpendicular bisector of QR is vertical, these lines will never intersect.
However, the perpendicular bisectors of RP and PQ do intersect. The perpendicular bisector of RP has a slope of 2 and passes through the midpoint (6, 9/2). The equation of this line is y - (9/2) = 2(x - 6).
Solving this equation, we can find the x-coordinate of the orthocenter by substituting the y-coordinate as 2 into the equation:
2 - (9/2) = 2(x - 6)
4 - 9 = 4x - 24
-5 = 4x - 24
4x = 19
x = 19/4
Plugging this x-coordinate back into the equation of the perpendicular bisector, we find the y-coordinate:
y - (9/2) = 2(19/4 - 6)
y - (9/2) = 19/2 - 12
y - (9/2) = 7/2
y = 7/2 + 9/2
y = 16/2
y = 8
Therefore, the orthocenter of triangle PQR is (19/4, 8).