If PL, QM and RN are the altitudes of triangle PQR whose orthocenter is O, then Q is the orthocenter of the triangle?
OPR
Understanding the Orthocenter of a Triangle
The orthocenter of a triangle is a fundamental concept in geometry. It is defined as the point where the three altitudes of the triangle intersect.
- An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or the line containing the opposite side).
In the given problem, we have triangle PQR, and its altitudes are PL, QM, and RN. These altitudes intersect at point O, which is the orthocenter of triangle PQR. This means:
- PL is perpendicular to QR.
- QM is perpendicular to PR.
- RN is perpendicular to PQ.
- PL, QM, and RN all pass through O.
Identifying the Orthocenter of Other Triangles
A key property related to the orthocenter of a triangle states that if O is the orthocenter of ▵PQR, then the vertices P, Q, and R are the orthocenters of the triangles formed by O and the other two vertices.
- P is the orthocenter of ▵OQR.
- Q is the orthocenter of ▵OPR.
- R is the orthocenter of ▵OPQ.
Let's verify this property for the triangle ▵OPR and show that Q is its orthocenter.
Verifying Q as the Orthocenter of Triangle OPR
For Q to be the orthocenter of ▵OPR, the altitudes of ▵OPR must intersect at Q. The altitudes of ▵OPR are the perpendiculars from its vertices O, P, and R to the opposite sides PR, OR, and OP, respectively.
Altitude from O to PR in ▵OPR
In the original triangle PQR, QM is the altitude from Q to PR. This means QM is perpendicular to PR. Since O is the orthocenter of PQR, O lies on the altitude QM. Therefore, the line containing O and perpendicular to PR is the line QM. This line passes through Q (as it is the altitude from Q in PQR).
Altitude from P to OR in ▵OPR
OR lies on the line segment RN, which is the altitude from R in ▵PQR. So, RN is perpendicular to PQ. This means the line PQ is perpendicular to RN (and thus perpendicular to OR). The line segment from vertex P of ▵OPR perpendicular to the opposite side OR is the line PQ. This line PQ clearly passes through Q.
Altitude from R to OP in ▵OPR
OP lies on the line segment PL, which is the altitude from P in ▵PQR. So, PL is perpendicular to QR. This means the line QR is perpendicular to PL (and thus perpendicular to OP). The line segment from vertex R of ▵OPR perpendicular to the opposite side OP is the line QR. This line QR clearly passes through Q.
Conclusion for ▵OPR
We have found that the altitudes of ▵OPR are the lines QM, PQ, and QR. All these three lines intersect at the point Q.
Therefore, Q is indeed the orthocenter of ▵OPR.
Summary of Orthocenter Relationships
| Triangle | Orthocenter |
|---|---|
| PQR | O |
| OQR | P |
| OPR | Q |
| OPQ | R |
Based on the property and verification, Q is the orthocenter of the triangle OPR.
Revision Table: Key Geometry Terms
| Term | Definition |
|---|---|
| Altitude | A line segment from a vertex perpendicular to the opposite side. |
| Orthocenter | The point of intersection of the three altitudes of a triangle. |
Additional Information: Properties of Orthocenter
The location of the orthocenter depends on the type of triangle:
- In an acute-angled triangle, the orthocenter lies inside the triangle.
- In a right-angled triangle, the orthocenter lies at the vertex containing the right angle.
- In an obtuse-angled triangle, the orthocenter lies outside the triangle.
The set of points {P, Q, R, O} is an orthocentric system. In such a system, any one point is the orthocenter of the triangle formed by the other three points.
Similar Questions
What is the area of quadrilateral ABCD?
ABC is a triangle right angled at B. Let D be the midpoint on AC. If BD = 6.5 cm, then what is AB 2 + BC 2 equal to?
AD is the median of the triangle ABC. If P is any point on AD, then which one of the following is correct?
In a triangle ABC, if 2 ∠A = 3 ∠B = 6 ∠C, then what is ∠A + ∠C equal to?
Consider the following statements :
1. The sum of any two sides of a triangle is less than twice the median drawn to the third side.
2. The perimeter of a triangle is greater than the sum of the three medians.
Which of the above statements is/are correct?
In a triangle, values of all the angles are integers (in degree measure). Which one of the following cannot be the proportion of their measures?
ABC is an equilateral triangle. The side BC is trisected at D such that BC = 3 BD. What is the ratio of AD 2to AB 2?
Two isosceles triangles have equal vertical angles and their areas are in the ratio 4.84 ∶ 5.29. What is the ratio of their corresponding heights?
Δ ABC is similar to Δ DEF. The perimeters of Δ ABC and Δ DEF are 40 cm and 30 cm respectively. What is the ratio of (BC + CA) to (EF + FD) equal to?
ABC is a triangle right angled at C. Let p be the length of the perpendicular drawn from C on AB. If BC = 6 cm and CA = 8 cm, then what is the value of p?
Important Questions from Triangles, Congruence and Similarity
G is the centroid of the equilateral triangle ABC. If AB = 8√ 3 cm, then the length of AG is equal to:
If sides of a triangle are 12 cm, 15 cm and 21 cm, then what is the inradius (in cm) of the triangle?
What is the area of quadrilateral ABCD?
It is given that ΔABC ~ ΔXYZ and Area ΔABC : Area ΔXYZ = 81 : 25. If AB = 18 cm, BC = 10 cm, CA = 15 cm, then what is the side XZ (in cm)?
Sides of two similar triangles are in the ratio 4 ∶ 9. Area of these triangles are in the ratio:
