Altitudes and the Orthocenter Basics

Grade 10th

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1) In △ABC, altitude AD is drawn to side BC. Which fact justifies that ∠ADB is a right angle?

Vertical angles theorem

Corresponding angles theorem

Definition of perpendicular lines

Triangle sum theorem

The altitude AD is defined as a perpendicular segment drawn from vertex A to side BC, which means that angle ADB is a right angle by definition.

Explanation

The altitude AD is defined as a perpendicular segment drawn from vertex A to side BC, which means that angle ADB is a right angle by definition.

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About This Quiz

Altitudes And The Orthocenter Basics - Quiz

Get ready to explore altitudes — lines that drop straight down from a vertex at a right angle to the opposite side. In this quiz, you’ll practice identifying altitudes, finding where they meet, and seeing how their intersection forms the orthocenter. You’ll also learn how the orthocenter’s position changes in... see moreacute, right, and obtuse triangles. By the end, you’ll understand how altitudes make triangles stand tall! see less

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2) Which point of concurrency of the altitudes of a triangle may lie inside, outside, or on the triangle depending on its type?

Centroid

Incenter

Circumcenter

Orthocenter

The orthocenter is the point where the altitudes of a triangle intersect. Its position varies depending on the type of triangle: it lies inside for acute triangles, at the vertex for right triangles, and outside for obtuse triangles.

Explanation

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3) In coordinate geometry, if a side of a triangle lies on the x-axis, the altitude to that side must be

Vertical

Horizontal

Diagonal

Parallel to the x-axis

In geometry, the altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. When a side lies on the x-axis, the altitude must be vertical to intersect the x-axis at a right angle.

Explanation

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4) In XYZ, altitudes from each vertex are drawn. What point do they meet at?

Centroid

Incenter

Circumcenter

Orthocenter

The point where the altitudes of a triangle intersect is called the Orthocenter. This point can be inside, outside, or on the triangle depending on the type of triangle.

Explanation

The point where the altitudes of a triangle intersect is called the Orthocenter. This point can be inside, outside, or on the triangle depending on the type of triangle.

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5) Which triangle will have its orthocenter outside the triangle?

Equilateral

Acute

Obtuse

Right

In an obtuse triangle, the orthocenter lies outside the triangle because the altitude from the vertex of the obtuse angle extends outside the triangle, unlike in acute and right triangles where the orthocenter is within the triangle.

Explanation

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6) The orthocenter of an equilateral triangle coincides with the ________.

Centroid

Circumcenter

Incenter

All of the above

In an equilateral triangle, all the special points - orthocenter, centroid, circumcenter, and incenter - coincide at the same point due to the triangle's symmetry.

Explanation

In an equilateral triangle, all the special points - orthocenter, centroid, circumcenter, and incenter - coincide at the same point due to the triangle's symmetry.

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7) The altitudes of a triangle are always

Concurrent

Equal in length

Perpendicular bisectors

Medians

The altitudes of a triangle are concurrent, meaning they all intersect at a single point known as the orthocenter. This is a defining property of altitudes in triangles.

Explanation

The altitudes of a triangle are concurrent, meaning they all intersect at a single point known as the orthocenter. This is a defining property of altitudes in triangles.

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8) An altitude of a triangle is a segment drawn from a vertex:

To the opposite side’s midpoint

Perpendicular to the opposite side

To the centroid

Bisecting the angle

An altitude in a triangle is defined as a line segment that extends from a vertex and is perpendicular to the line containing the opposite side, thus creating a right angle with that side.

Explanation

An altitude in a triangle is defined as a line segment that extends from a vertex and is perpendicular to the line containing the opposite side, thus creating a right angle with that side.

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9) The three altitudes of a triangle are concurrent at the:

Centroid

Orthocenter

Incenter

Circumcenter

The orthocenter is the point where the three altitudes of a triangle intersect. Each altitude is a line segment from a vertex perpendicular to the opposite side, and their intersection point is known as the orthocenter.

Explanation

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10) The orthocenter of an acute triangle lies:

Inside the triangle

Outside the triangle

On a vertex

On a side

In an acute triangle, all three angles are less than 90 degrees, and the orthocenter, which is the point where the altitudes intersect, is located within the triangle.

Explanation

In an acute triangle, all three angles are less than 90 degrees, and the orthocenter, which is the point where the altitudes intersect, is located within the triangle.

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11) The orthocenter of an obtuse triangle lies:

Inside the triangle

Outside the triangle

At the centroid

At the incenter

In an obtuse triangle, the orthocenter, which is the intersection point of the altitudes, is located outside the triangle because one of the angles exceeds 90 degrees.

Explanation

In an obtuse triangle, the orthocenter, which is the intersection point of the altitudes, is located outside the triangle because one of the angles exceeds 90 degrees.

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12) The orthocenter of a right triangle lies:

Inside, but not on a vertex

At the centroid

On the hypotenuse

At the vertex of the right angle

In a right triangle, the orthocenter is located at the vertex of the right angle because the altitudes from the other two vertices meet at this point.

Explanation

In a right triangle, the orthocenter is located at the vertex of the right angle because the altitudes from the other two vertices meet at this point.

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13) Which triangle has its orthocenter, centroid, and circumcenter all at the same point?

Scalene

Isosceles

Right

Equilateral

In an equilateral triangle, all three points—orthocenter, centroid, and circumcenter—coincide at the same point due to the symmetry and equal angles of the triangle.

Explanation

In an equilateral triangle, all three points—orthocenter, centroid, and circumcenter—coincide at the same point due to the symmetry and equal angles of the triangle.

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14) The altitude from vertex A in triangle ABC is drawn to side BC. Which condition must hold?

AD ⊥ BC

AD bisects ∠A

D is the midpoint of BC

AD bisects ∠B

In a triangle, the altitude is a line segment from a vertex that is perpendicular to the opposite side. Thus, for the altitude from vertex A to be valid, it must form a right angle with side BC, meaning that AD must be perpendicular to BC.

Explanation

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15) If an altitude of a triangle has length 9, and the distance from the orthocenter to that vertex is 3, then the distance from the orthocenter to the opposite side is:

In a triangle, the orthocenter is the point where the altitudes intersect. The distance from the orthocenter to the vertex is one part of the total altitude measurement. Since the total altitude is 9 and the distance from the orthocenter to the vertex is 3, the remaining distance to the opposite side must be 9 - 3 = 6.

Explanation

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16) In a triangle, each altitude forms a right angle with the side it meets. Which theorem about lines and angles justifies that this angle is a right angle?

Vertical Angles Theorem

Corresponding Angles Theorem

Perpendicular Bisector Theorem

Definition of Perpendicular Lines

The Definition of Perpendicular Lines states that if two lines intersect to form a right angle, they are perpendicular to each other. Since the altitude meets the side of the triangle at a right angle, it confirms that the altitude is perpendicular to that side.

Explanation

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17) The orthocenter of triangle ABC has coordinates (0,0). If A = (0,0), what type of triangle is this?

Right triangle

Equilateral triangle

Scalene acute triangle

Obtuse triangle

A triangle can only be a right triangle if one of its vertices is at the orthocenter while the other two vertices create a right angle with respect to the orthocenter. Since point A is the orthocenter, it indicates that triangle ABC forms a right angle at one of the other vertices.

Explanation

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18) In △PQR, altitude QS is drawn to side PR. Which angle pair is guaranteed to be congruent because of this construction?

∠QSP and ∠QSR

∠QSP and ∠PRS

∠PQS and ∠QRS

∠PQR and ∠PSQ

When an altitude is drawn from a vertex to the opposite side in a triangle, it creates two right triangles. The angles formed at the point where the altitude meets the base are congruent due to the property of vertical angles, and the two angles QSP and QSR are corresponding angles formed by the altitude QS to side PR.

Explanation

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19) If △ABC has altitudes of lengths 6, 8, and 10, which of the following is true?

They meet at the incenter

They meet at the centroid

They meet at the orthocenter

They do not meet at one point

The altitudes of a triangle intersect at a single point called the orthocenter. Thus, option C is correct as it states that the altitudes meet at the orthocenter.

Explanation

The altitudes of a triangle intersect at a single point called the orthocenter. Thus, option C is correct as it states that the altitudes meet at the orthocenter.

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20) Which statement is false?

Every triangle has three altitudes

Altitudes are always concurrent

The orthocenter always lies inside the triangle

Altitudes are perpendicular to the opposite side

The orthocenter of a triangle can lie outside the triangle, especially in obtuse triangles, making statement C false.

Explanation

The orthocenter of a triangle can lie outside the triangle, especially in obtuse triangles, making statement C false.

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Oct 12, 2025

Quiz Edited by
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Sep 19, 2025

Quiz Created by
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In △ABC, altitude AD is drawn to side BC. Which fact justifies that...

Which point of concurrency of the altitudes of a triangle may lie...

In coordinate geometry, if a side of a triangle lies on the x-axis,...

In XYZ, altitudes from each vertex are drawn. What point do they meet...

Which triangle will have its orthocenter outside the triangle?

The orthocenter of an equilateral triangle coincides with the...

The altitudes of a triangle are always

An altitude of a triangle is a segment drawn from a vertex:

The three altitudes of a triangle are concurrent at the:

The orthocenter of an acute triangle lies:

The orthocenter of an obtuse triangle lies:

The orthocenter of a right triangle lies:

Which triangle has its orthocenter, centroid, and circumcenter all at...

The altitude from vertex A in triangle ABC is drawn to side BC. Which...

If an altitude of a triangle has length 9, and the distance from the...

In a triangle, each altitude forms a right angle with the side it...

The orthocenter of triangle ABC has coordinates (0,0). If A = (0,0),...

In △PQR, altitude QS is drawn to side PR. Which angle pair is...

If △ABC has altitudes of lengths 6, 8, and 10, which of the...

Which statement is false?

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