Altitudes And Orthocenter: Proof Applications

1) Which of the following best describes the difference between an altitude and a median?

A median is always longer:

An altitude is always perpendicular, but a median may not be:

A median always lies outside the triangle, but an altitude does not:

An altitude bisects the angle, but a median does not:

A median connects a vertex to the midpoint of the opposite side, while an altitude must be perpendicular.

Explanation

A median connects a vertex to the midpoint of the opposite side, while an altitude must be perpendicular.

2) The orthocenter of △XYZ is located at vertex X. Which statement must be true?

△XYZ is equilateral.

△XYZ is a right triangle with ∠X as the right angle.

△XYZ is isosceles with base YZ.

△XYZ is obtuse with ∠X obtuse.

In a right triangle, the orthocenter is at the vertex where the right angle occurs.

Explanation

In a right triangle, the orthocenter is at the vertex where the right angle occurs.

3) The point where the three altitudes of a triangle meet is called the:

Centroid

Orthocenter

Incenter

Circumcenter

In an acute triangle, all angles are less than 90°, so the orthocenter lies inside the triangle.

Explanation

In an acute triangle, all angles are less than 90°, so the orthocenter lies inside the triangle.

4) In △PQR, altitude PS is drawn to side QR. If ∠P = 40° and ∠Q = 60°, what is ∠PSQ?

30°

40°

50°

90°

An altitude forms a 90° angle with the side it intersects.

The other angles don’t affect this property.

Explanation

An altitude forms a 90° angle with the side it intersects.

The other angles don’t affect this property.

5) If the altitudes of △ABC are constructed, they all intersect at a single point called the:

Centroid

Orthocenter

Incenter

Circumcenter

The three altitudes always intersect at a single point known as the orthocenter.

Explanation

The three altitudes always intersect at a single point known as the orthocenter.

6) In △PQR, altitude QS is drawn to side PR. Which angle is guaranteed to be a right angle?

∠PQS

∠QSP

∠PSQ

∠PQR

Altitudes are always perpendicular to the side they meet, forming a right angle.

Explanation

Altitudes are always perpendicular to the side they meet, forming a right angle.

7) An altitude of a triangle is:

A line through a vertex that meets the opposite side at a right angle.

A line through a vertex and the opposite side’s midpoint.

A line through a vertex that bisects an angle.

A line segment that is always the longest side.

An altitude is a line segment from a vertex that is perpendicular to the opposite side. It shows height in a triangle.

Explanation

An altitude is a line segment from a vertex that is perpendicular to the opposite side. It shows height in a triangle.

8) In triangle XYZ, if the orthocenter lies inside the triangle, what type of triangle is XYZ?

Acute

Right

Obtuse

Isosceles

In an acute triangle, all angles are less than 90°, so the orthocenter lies inside the triangle.

Explanation

In an acute triangle, all angles are less than 90°, so the orthocenter lies inside the triangle.

9) In triangle PQR, if the orthocenter is located at vertex Q, what type of triangle is PQR?

Acute

Right

Obtuse

Equilateral

In a right triangle, the orthocenter is at the vertex where the right angle is located.

Explanation

In a right triangle, the orthocenter is at the vertex where the right angle is located.

10) In triangle ABC, altitude AD is drawn to side BC. Which of the following is true about point D?

D is the midpoint of BC

D lies on side BC

D is the centroid of triangle ABC

D is the circumcenter of triangle ABC

The foot of the altitude always lies on the side to which it is drawn.

Explanation

The foot of the altitude always lies on the side to which it is drawn.

11) Which triangle has all its centers (centroid, orthocenter, circumcenter, incenter) at the same point?

Scalene

Isosceles

Right

Equilateral

In an equilateral triangle, all centers—centroid, orthocenter, circumcenter, and incenter—coincide at one point.

Explanation

In an equilateral triangle, all centers—centroid, orthocenter, circumcenter, and incenter—coincide at one point.

12) Which of the following is not always true?

Each triangle has three altitudes:

Altitudes intersect at a single point:

The orthocenter may fall inside, outside, or on the triangle:

The orthocenter is not always located at the centroid:

The orthocenter and centroid are generally different points except in equilateral triangles.

Explanation

The orthocenter and centroid are generally different points except in equilateral triangles.

13) In △DEF, the vertices are D(2,2), E(8,2), and F(2,10). Find the coordinates of the orthocenter.

(2,2)

(2,10)

(8,2)

(2,8)

Here, D(2,2) is the right-angle vertex because sides DE and DF are perpendicular.

Thus, the orthocenter is at D(2,2).

Explanation

Here, D(2,2) is the right-angle vertex because sides DE and DF are perpendicular.

Thus, the orthocenter is at D(2,2).

14) Which is always true about altitudes?

They divide the opposite side into equal halves:

They are angle bisectors:

They are concurrent:

They are equal in length:

All altitudes in a triangle meet at one point—the orthocenter.

That’s what makes them concurrent.

Explanation

All altitudes in a triangle meet at one point—the orthocenter.

That’s what makes them concurrent.

15) In triangle ABC, the altitudes intersect at point H. If triangle ABC is equilateral with side length 12, what is the distance from the centroid to the orthocenter?

0

4

6

12

In an equilateral triangle, all centers coincide.

The centroid and orthocenter are at the same point, so the distance is 0.

Explanation

In an equilateral triangle, all centers coincide.

The centroid and orthocenter are at the same point, so the distance is 0.

16) In triangle ABC, the slopes of two altitudes are -3 and 1/2. What can you conclude about the triangle?

It must be isosceles:

It must be right:

It is equilateral:

It is scalene but not right:

Slopes that are negative reciprocals (−3 and 1/3 or −2 and 1/2) mean the lines are perpendicular, forming a right triangle.

Explanation

Slopes that are negative reciprocals (−3 and 1/3 or −2 and 1/2) mean the lines are perpendicular, forming a right triangle.

17) Which statement is always true?

The orthocenter is the balance point.

The orthocenter always lies inside the triangle.

Altitudes intersect at the orthocenter.

The orthocenter is equidistant from the vertices.

No matter the triangle type, all altitudes always meet at the orthocenter—that’s their defining property.

Explanation

No matter the triangle type, all altitudes always meet at the orthocenter—that’s their defining property.

18) In △LMN, the orthocenter is located outside the triangle. What type of triangle must △LMN be?

Acute

Right

Obtuse

Equilateral

For an obtuse triangle, one angle is greater than 90°, causing the orthocenter to fall outside the triangle.

Explanation

For an obtuse triangle, one angle is greater than 90°, causing the orthocenter to fall outside the triangle.

19) Which point of concurrency may lie inside, outside, or on the triangle depending on the type?

Centroid

Incenter

Orthocenter

Circumcenter

The orthocenter changes location:

Inside → acute triangle

On → right triangle

Outside → obtuse triangle

Explanation

The orthocenter changes location:

Inside → acute triangle

On → right triangle

Outside → obtuse triangle

20) If △ABC has altitudes AD, BE, and CF, they intersect at:

Centroid

Incenter

Orthocenter

Circumcenter

No matter where they are drawn, altitudes always meet at the orthocenter.

Explanation

No matter where they are drawn, altitudes always meet at the orthocenter.

Altitudes and Orthocenter: Proof Applications