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7th Edition

Alexander + 2 others

Publisher: Cengage,

ISBN: 9781337614085

Chapter 7.2, Problem 10E

Textbook Problem

a) To locate the orthocenter, is it necessary to construct all three altitude of a right triangle?

b) What point is the orthocenter of any right triangle?

Expert Solution

To determine

(a)

**To find:**

To locate the orthocenter, is it necessary to construct all three altitudes of a right triangle.

**Definition:**

Orthocenter:

The point where the three altitudes of a triangle meet. An altitude is a line that goes through vertex (corner point) and is at right angles to the opposite side.

**Given:**

To locate the orthocenter, is it necessary to construct all three altitudes of a right triangle.

**Calculation:**

By the above definition,

The orthocenter is the point where all three altitudes of the triangle intersect. An altitudes is a line which passes through a vertex of a triangle, and is perpendicular to the opposite side.

The three altitudes all intersect at the same point so we only need two altitude to locate it.

For example:

In a right angle triangle, two sides are perpendicular to each other. The legs of the right triangle are two of the altitudes.

Expert Solution

To determine

(b)

**To find:**

What point is the orthocenter of any right triangle.

Elementary Geometry For College Students, 7e

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Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - In Exercises 3 to 8, use the drawing provided....Ch. 7.1 - In Exercises 3 to 8, use the drawing provided....Ch. 7.1 - In Exercises 3 to 8, use the drawing provided....Ch. 7.1 - In Exercises 3 to 8, use the drawing provided....Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...

Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - In Exercises 11 to 22, sketch and describe each...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - In Exercises 39 and 42, refer to the line segments...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - In Exercises 39 and 42, refer to the line segments...Ch. 7.1 - Construct the three angle bisectors and then the...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - In Exercises 39 and 42, refer to the line segments...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.1 - In Exercises 39 and 42, refer to the line segments...Ch. 7.1 - Note: Exercises preceded by an asterisk are of a...Ch. 7.2 - Note: Exercise preceded by an asterisk are of a...Ch. 7.2 - Note: Exercises preceded by an asterisk are of a...Ch. 7.2 - Note: Exercises preceded by an asterisk are of a...Ch. 7.2 - a What is the general name of the point of...Ch. 7.2 - a What is the general name of the point of...Ch. 7.2 - Which lines or line segments or rays must be drawn...Ch. 7.2 - Which lines or line segments or rays must be drawn...Ch. 7.2 - a Is it really necessary to construct all three...Ch. 7.2 - Which lines, line segments, or rays are used to...Ch. 7.2 - a To locate the orthocenter, is it necessary to...Ch. 7.2 - To locate the centroid of a triangle, is it...Ch. 7.2 - For what type of triangle are the angle bisectors,...Ch. 7.2 - What point on a right triangle is the circumcenter...Ch. 7.2 - Must the centroid of an isosceles triangle lie on...Ch. 7.2 - Draw a triangle and, by construction, find its...Ch. 7.2 - Draw an acute triangle and, by construction, find...Ch. 7.2 - Draw an obtuse triangle and, by construction, find...Ch. 7.2 - Draw an acute triangle and, by construction, find...Ch. 7.2 - Draw an obtuse triangle and, by construction, find...Ch. 7.2 - Draw an acute triangle and, by construction, find...Ch. 7.2 - Draw an obtuse triangle and, by construction, find...Ch. 7.2 - Is the incenter always located in the interior of...Ch. 7.2 - Is the circumcenter always located in the interior...Ch. 7.2 - Find the length of the radius of the inscribed...Ch. 7.2 - Find the distance from the circumcenter to each...Ch. 7.2 - A triangle has angles measuring 30, 30, and 120....Ch. 7.2 - In MNP, medians MB, NA and PC intersect at...Ch. 7.2 - In MNP, medians MB, NA and PC intersect at...Ch. 7.2 - Given: Isosceles RST RS=RT=17andST=16. Medians RZ,...Ch. 7.2 - Given: Isosceles RST RS=RT=10andST=16. Medians RZ,...Ch. 7.2 - Draw a triangle. Construct its inscribed circle.Ch. 7.2 - Draw a triangle. Construct its circumscribed...Ch. 7.2 - For what type of triangle will be the incenter and...Ch. 7.2 - Does a rectangle have a an incenter? b a...Ch. 7.2 - Does a square have a an incenter? b a...Ch. 7.2 - Does a regular pentagon have a an incenter?.b a...Ch. 7.2 - Does a rhombus have a an incenter? b a...Ch. 7.2 - Does a isosceles trapezoid have a an incenter?.b a...Ch. 7.2 - In ABC, the bisectors of the angle are concurrent...Ch. 7.2 - In ABC, the altitudes are concurrent at point D....Ch. 7.2 - A distributing company plans an Illinois location...Ch. 7.2 - There are plans to locate a disaster response...Ch. 7.2 - A circle is inscribed in an isosceles triangle...Ch. 7.2 - For exercises 44 to 46, O is the inscribed circle...Ch. 7.2 - For exercises 44 to 46, O is the inscribed circle...Ch. 7.2 - For exercises 44 to 46, O is the inscribed circle...Ch. 7.3 - Describe, if possible, how you would inscribe a...Ch. 7.3 - What condition must be satisfied for it to be...Ch. 7.3 - Describe, if possible, how you would inscribe a...Ch. 7.3 - What condition must be satisfied for it to be...Ch. 7.3 - In Exercises 5 to 8, perform constructions....Ch. 7.3 - In Exercises 5 to 8, perform constructions....Ch. 7.3 - In Exercises 5 to 8, perform constructions....Ch. 7.3 - In Exercises 5 to 8, perform constructions....Ch. 7.3 - Find the perimeter of a regular octagon if the...Ch. 7.3 - In a regular polygon with each side of length 6.5...Ch. 7.3 - If the perimeter of a regular dodecagon 12 sides...Ch. 7.3 - If the apothem of a square measures 5 cm, find the...Ch. 7.3 - Find the lengths of the apothem and the radius of...Ch. 7.3 - Find the lengths of the apothem and the radius of...Ch. 7.3 - Find the lengths of the side and the radius of an...Ch. 7.3 - Find the lengths of the side and the radius of a...Ch. 7.3 - Find the measure of a central angle of a regular...Ch. 7.3 - Find the measure of a central angle of a regular...Ch. 7.3 - Find the number of sides of a regular polygon that...Ch. 7.3 - Find the number of sides of a regular polygon that...Ch. 7.3 - Find the measure of each interior angle of a...Ch. 7.3 - Find the measure of each interior angle of a...Ch. 7.3 - Find the measure of each exterior angle of a...Ch. 7.3 - Find the measure of each exterior angle of a...Ch. 7.3 - Find the number of sides for a regular polygon in...Ch. 7.3 - Find the number of sides for a regular polygon in...Ch. 7.3 - Is there a regular polygon for which each central...Ch. 7.3 - If possible, draw an equilateral and concave: a...Ch. 7.3 - If possible, draw an equilateral and concave: a...Ch. 7.3 - Given regular hexagon ABCDEF with each side of...Ch. 7.3 - Given regular octagon RSTUVWXY with each side of...Ch. 7.3 - Given that RSTVQ is a regular pentagon and PQR is...Ch. 7.3 - Given: Regular pentagon RSTVQ with equilateral ...Ch. 7.3 - Given: Regular pentagon JKLMN not shown with...Ch. 7.3 - Is there a regular polygon with 8 diagonals? If...Ch. 7.3 - Is there a regular polygon with 12 diagonals? If...Ch. 7.3 - Find the measure of a central angle of a regular...Ch. 7.3 - Find the measure of a central angle of a regular...Ch. 7.3 - A regular polygon of n sides has an exterior angle...Ch. 7.3 - A regular polygon of n sides has an exterior angle...Ch. 7.3 - A regular polygon of n sides has an interior angle...Ch. 7.3 - A regular polygon of n sides has an interior angle...Ch. 7.CR - In Review Exercises 1 to 6, use the figure shown....Ch. 7.CR - In Review Exercises 1 to 6, use the figure shown....Ch. 7.CR - In Review Exercises 1 to 6, use the figure shown....Ch. 7.CR - In Review Exercises 1 to 6, use the figure shown....Ch. 7.CR - In Review Exercises 1 to 6, use the figure shown....Ch. 7.CR - In Review Exercises 1 to 6, use the figure shown....Ch. 7.CR - In Review Exercises 7 to 13, sketch and describe...Ch. 7.CR - In Review Exercises 7 to 13, sketch and describe...Ch. 7.CR - In Review Exercises 7 to 13, sketch and describe...Ch. 7.CR - In Review Exercises 7 to 13, sketch and describe...Ch. 7.CR - In Review Exercises 7 to 13, sketch and describe...Ch. 7.CR - In Review Exercise 7 to 13, sketch and describe...Ch. 7.CR - In Review Exercises 7 to 13, sketch and describe...Ch. 7.CR - In Review Exercises 14 to 17, sketch and describe...Ch. 7.CR - In Review Exercises 14 to 17, sketch and describe...Ch. 7.CR - In Review Exercises 14 to 17, sketch and describe...Ch. 7.CR - In Review Exercises 14 to 17, sketch and describe...Ch. 7.CR - In Review Exercises 18 to 23, use construction...Ch. 7.CR - In Review Exercises 18 to 23, use construction...Ch. 7.CR - In Review Exercises 18 to 23, use construction...Ch. 7.CR - In Review Exercises 18 to 23, use construction...Ch. 7.CR - In Review Exercises 18 to 23, use construction...Ch. 7.CR - In Review Exercises 18 to 23, use construction...Ch. 7.CR - Given: ABC with medians AE,CD,BF Find: a BG if...Ch. 7.CR - Given: ABC in Exercise 24 with medians AE,CD,BF...Ch. 7.CR - For a regular pentagon, find the measure of each a...Ch. 7.CR - For a regular decagon 10 sides, find the measure...Ch. 7.CR - In a regular polygon, each central angle measures...Ch. 7.CR - In a regular polygon, the apothem measures 3 in....Ch. 7.CR - Can a circle be circumscribed about each of the...Ch. 7.CR - Can a circle be inscribed in each of the following...Ch. 7.CR - The length of the radius of a circle inscribed in...Ch. 7.CR - The length of the radius of a circle inscribed in...Ch. 7.CR - Sketch the parabola that has directrix d and focus...Ch. 7.CT - Draw and describe the locus of points in the plane...Ch. 7.CT - Draw and describe the locus of points in the plane...Ch. 7.CT - Draw and describe the locus of points in the plane...Ch. 7.CT - Describe the locus of points in a plane that are...Ch. 7.CT - Describe the locus of points in space that are at...Ch. 7.CT - For a given triangle such as ABC, what word...Ch. 7.CT - For a given triangle such as ABC, what word...Ch. 7.CT - In what type of triangle are the angle bisectors,...Ch. 7.CT - Which of the following must be concurrent at an...Ch. 7.CT - Classify as true/false: aA circle can be inscribed...Ch. 7.CT - An equilateral triangle has a radius of length 3...Ch. 7.CT - For a regular pentagon, find the measure of each...Ch. 7.CT - The measure of each central angle of a regular...Ch. 7.CT - For a regular octagon, the length of the apothem...Ch. 7.CT - For a regular hexagon ABCDEF, the length of side...Ch. 7.CT - For rectangle MNPQ, points A, B, C and D are the...Ch. 7.CT - Sketch the parabola that has directrix d and focus...

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