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:

An altitude is a line segment from a vertex to the opposite side, forming a right angle with that side, while a median connects a vertex to the midpoint of the opposite side and does not have to be perpendicular.

Explanation

An altitude is a line segment from a vertex to the opposite side, forming a right angle with that side, while a median connects a vertex to the midpoint of the opposite side and does not have to 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 any triangle, if the orthocenter is located at one of the vertices, that vertex must be a right angle. Therefore, triangle XYZ must be a right triangle with angle X being the right angle.

Explanation

In any triangle, if the orthocenter is located at one of the vertices, that vertex must be a right angle. Therefore, triangle XYZ must be a right triangle with angle X being the right angle.

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

30°

40°

50°

90°

In triangle PQR, the sum of angles is always 180 degrees. Since angle P is 40 degrees and angle Q is 60 degrees, angle R can be calculated as 180 - (40 + 60) = 80 degrees. The altitude PS creates a right angle at point S, hence angle PSQ is 90 degrees.

Explanation

In triangle PQR, the sum of angles is always 180 degrees. Since angle P is 40 degrees and angle Q is 60 degrees, angle R can be calculated as 180 - (40 + 60) = 80 degrees. The altitude PS creates a right angle at point S, hence angle PSQ is 90 degrees.

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

Centroid

Orthocenter

Incenter

Circumcenter

The orthocenter is the point where the altitudes of a triangle intersect. This point can lie inside, on, or outside the triangle depending on the type of triangle (acute, right, or obtuse).

Explanation

The orthocenter is the point where the altitudes of a triangle intersect. This point can lie inside, on, or outside the triangle depending on the type of triangle (acute, right, or obtuse).

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

∠PQS

∠QSP

∠PSQ

∠PQR

In a triangle, an altitude is defined as a perpendicular segment from a vertex to the line containing the opposite side. Therefore, angle QSP is formed where the altitude QS meets side PR, ensuring it is a right angle.

Explanation

In a triangle, an altitude is defined as a perpendicular segment from a vertex to the line containing the opposite side. Therefore, angle QSP is formed where the altitude QS meets side PR, ensuring it is a right angle.

6) 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 in a triangle is defined as the perpendicular line segment drawn from a vertex to the opposite side, creating a right angle at the point where it intersects the side.

Explanation

An altitude in a triangle is defined as the perpendicular line segment drawn from a vertex to the opposite side, creating a right angle at the point where it intersects the side.

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 of a triangle is defined as a perpendicular line segment drawn from a vertex to the opposite side. This definition ensures that the altitude meets the side at a right angle, which is essential for calculating the area of the triangle.

Explanation

An altitude of a triangle is defined as a perpendicular line segment drawn from a vertex to the opposite side. This definition ensures that the altitude meets the side at a right angle, which is essential for calculating the area of the 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 degrees, and the orthocenter, which is the intersection of the altitudes, lies inside the triangle. In contrast, in a right triangle, the orthocenter is on the vertex of the right angle, and in an obtuse triangle, it lies outside.

Explanation

In an acute triangle, all angles are less than 90 degrees, and the orthocenter, which is the intersection of the altitudes, lies inside the triangle. In contrast, in a right triangle, the orthocenter is on the vertex of the right angle, and in an obtuse triangle, it lies outside.

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

Acute

Right

Obtuse

Equilateral

In a triangle, the orthocenter is the point where the altitudes intersect. If the orthocenter is at a vertex (in this case, vertex Q), it indicates that the angle at that vertex is a right angle, making triangle PQR a right triangle.

Explanation

In a triangle, the orthocenter is the point where the altitudes intersect. If the orthocenter is at a vertex (in this case, vertex Q), it indicates that the angle at that vertex is a right angle, making triangle PQR a right triangle.

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

Point D is where the altitude from vertex A meets side BC, meaning it lies directly on side BC.

Explanation

Point D is where the altitude from vertex A meets side BC, meaning it lies directly on side BC.

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

Scalene

Isosceles

Right

Equilateral

An equilateral triangle is unique in that all its special centers coincide at a single point, due to its equal angles and sides, making it symmetric.

Explanation

An equilateral triangle is unique in that all its special centers coincide at a single point, due to its equal angles and sides, making it symmetric.

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 of a triangle is the point where the three altitudes intersect. However, it can be located at various positions depending on the type of triangle: inside for acute triangles, outside for obtuse triangles, and at the vertex for right triangles. In contrast, the centroid is the point of intersection of the medians and is always located inside the triangle.

Explanation

The orthocenter of a triangle is the point where the three altitudes intersect. However, it can be located at various positions depending on the type of triangle: inside for acute triangles, outside for obtuse triangles, and at the vertex for right triangles. In contrast, the centroid is the point of intersection of the medians and is always located inside the triangle.

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)

The orthocenter of a triangle is the point where the three altitudes intersect. In triangle DEF, the altitudes from vertices D and E will meet at point D since it lies on the line connecting F and the midpoint of DE, which is the intersection point at (2,2).

Explanation

The orthocenter of a triangle is the point where the three altitudes intersect. In triangle DEF, the altitudes from vertices D and E will meet at point D since it lies on the line connecting F and the midpoint of DE, which is the intersection point at (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:

Altitudes of a triangle are the perpendicular segments from each vertex to the line containing the opposite side, and they meet at a single point called the orthocenter, making them concurrent.

Explanation

Altitudes of a triangle are the perpendicular segments from each vertex to the line containing the opposite side, and they meet at a single point called the orthocenter, making 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, the centroid and orthocenter coincide, meaning the distance between them is zero.

Explanation

In an equilateral triangle, the centroid and orthocenter coincide, meaning the distance between them is zero.

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:

In a triangle, if two altitudes have slopes that are negative and positive reciprocals, it indicates the triangle is right-angled. The slope of one altitude being -3 implies that the triangle has a right angle opposite to the side for which this altitude is drawn.

Explanation

In a triangle, if two altitudes have slopes that are negative and positive reciprocals, it indicates the triangle is right-angled. The slope of one altitude being -3 implies that the triangle has a right angle opposite to the side for which this altitude is drawn.

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:

The orthocenter is the point where the three altitudes of a triangle intersect, making statement C always true. Other statements may not hold for all types of triangles.

Explanation

The orthocenter is the point where the three altitudes of a triangle intersect, making statement C always true. Other statements may not hold for all types of triangles.

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

Acute

Right

Obtuse

Equilateral

In an obtuse triangle, one of the angles is greater than 90 degrees. The orthocenter, which is the point where the altitudes of the triangle intersect, lies outside the triangle in this case, which is a defining characteristic of obtuse triangles.

Explanation

In an obtuse triangle, one of the angles is greater than 90 degrees. The orthocenter, which is the point where the altitudes of the triangle intersect, lies outside the triangle in this case, which is a defining characteristic of obtuse triangles.

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

Centroid

Incenter

Orthocenter

Circumcenter

The orthocenter of a triangle is the point where the altitudes intersect. Depending on the type of triangle (acute, right, or obtuse), the orthocenter can be located inside, on, or outside the triangle.

Explanation

The orthocenter of a triangle is the point where the altitudes intersect. Depending on the type of triangle (acute, right, or obtuse), the orthocenter can be located inside, on, or outside the triangle.

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

Centroid

Incenter

Orthocenter

Circumcenter

In a triangle, the altitudes are the perpendicular segments from each vertex to the opposite side. The point where the three altitudes intersect is called the orthocenter.

Explanation

In a triangle, the altitudes are the perpendicular segments from each vertex to the opposite side. The point where the three altitudes intersect is called the orthocenter.