Non-right triangles & trigonometry Flashcards
What is a non-right triangle?
A triangle that does not contain a right angle (90 degrees).
True or False: All triangles have at least one right angle.
False
What are the three types of non-right triangles based on their angles?
Acute, obtuse, and scalene triangles.
Fill in the blank: A triangle with all angles less than 90 degrees is called a __________ triangle.
acute
Fill in the blank: A triangle with one angle greater than 90 degrees is called a __________ triangle.
obtuse
What is the longest side of a triangle opposite the largest angle called?
The hypotenuse.
Define the Law of Sines.
The ratio of the length of a side of a triangle to the sine of its opposite angle is constant.
If a triangle has sides of lengths a, b, and c, and angles A, B, and C opposite those sides, what does the Law of Sines state?
a/sin(A) = b/sin(B) = c/sin(C)
True or False: The Law of Cosines can be used to find an unknown angle in a non-right triangle.
True
What is the formula for the Law of Cosines?
c² = a² + b² - 2ab*cos(C)
What does the term ‘scalene triangle’ refer to?
A triangle with all sides of different lengths.
What is the sum of the interior angles of any triangle?
180 degrees.
In a triangle, if one angle measures 120 degrees, what can you say about the other two angles?
They must sum to 60 degrees.
Fill in the blank: The area of a triangle can be calculated using the formula A = 1/2 * base * height. This is true for __________ triangles.
all
What is the formula for the area of a triangle using the Law of Sines?
A = (1/2)ab*sin(C)
True or False: The Law of Cosines can only be used for obtuse triangles.
False
What is a characteristic of an isosceles triangle?
It has at least two sides of equal length.
What is the relationship between the angles in an isosceles triangle?
The angles opposite the equal sides are also equal.
Fill in the blank: A triangle with all sides equal is called an __________ triangle.
equilateral
In an equilateral triangle, what is the measure of each angle?
60 degrees.
What is the circumradius of a triangle?
The radius of the circumcircle, which passes through all three vertices of the triangle.
What does the term ‘inradius’ refer to in triangle geometry?
The radius of the incircle, which touches all three sides of the triangle.
True or False: The circumradius can be calculated using R = a / (2 * sin(A)).
True
What is the formula for the area of a triangle using Heron’s formula?
A = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2.
Define s in the context of Heron’s formula.
The semi-perimeter of the triangle.
What is the relationship between the sides of a triangle and the angles according to the Law of Sines?
The larger the angle, the longer the opposite side.
Fill in the blank: In a triangle, if angle A is 30 degrees and angle B is 60 degrees, angle C must be __________ degrees.
90
What is the significance of the angle-side relationship in triangles?
It helps determine the length of sides based on the angles.
True or False: The angles of a triangle can be any value as long as they sum to 180 degrees.
True
What is a common application of the Law of Sines in real life?
Calculating distances in navigation and surveying.
Fill in the blank: The angle opposite the longest side of a triangle is the __________ angle.
largest
What is the relationship between the sides of a triangle according to the Triangle Inequality Theorem?
The sum of the lengths of any two sides must be greater than the length of the third side.
True or False: The Law of Cosines can be used to find the length of a side when two sides and the included angle are known.
True
What is the formula for finding an angle using the Law of Cosines?
C = cos⁻¹[(a² + b² - c²) / (2ab)]
Fill in the blank: A triangle with angles measuring 45, 45, and 90 degrees is called a __________ triangle.
45-45-90
What is the ratio of the sides in a 45-45-90 triangle?
1:1:√2
What is the ratio of the sides in a 30-60-90 triangle?
1:√3:2
True or False: The Law of Sines can be used to solve for an unknown side when two angles and one side are known.
True
What is the significance of the altitude in a triangle?
It helps determine the height and area of the triangle.
Fill in the blank: The __________ of a triangle is the segment from a vertex to the opposite side that is perpendicular to that side.
altitude
What is meant by an ‘ambiguous case’ in the context of the Law of Sines?
When two sides and a non-included angle are known, leading to potentially two different triangles.
True or False: The area of a triangle can be found using the base and height.
True
What is the formula for the circumradius R of a triangle with sides a, b, c?
R = (abc) / (4A), where A is the area of the triangle.
Fill in the blank: The __________ is the point where the three altitudes of a triangle intersect.
orthocenter
What does the term ‘centroid’ refer to in triangle geometry?
The point where the three medians of a triangle intersect.
What is the function of the median in a triangle?
It connects a vertex to the midpoint of the opposite side.
True or False: The circumcenter is the point where the three medians of a triangle intersect.
False
Fill in the blank: The __________ is the center of the circle that can be inscribed in a triangle.
incenter
What is the formula to calculate the inradius r of a triangle?
r = A/s, where A is the area and s is the semi-perimeter.
What is the relationship between the circumradius (R) and the inradius (r) in a triangle?
R ≥ 2r, meaning the circumradius is at least twice the inradius.
True or False: The sum of the lengths of the three sides of a triangle is equal to the perimeter.
True
What is the perimeter of a triangle with sides a, b, and c?
P = a + b + c.
Fill in the blank: The __________ of a triangle is a point that divides each median into a ratio of 2:1.
centroid
What is the relationship between angles and sides in a triangle according to the Triangle Inequality Theorem?
The largest angle is opposite the longest side.
True or False: The angles in a triangle can be equal to or greater than 180 degrees.
False
What is the area of a triangle with sides 5, 12, and 13 using Heron’s formula?
30 square units.
Fill in the blank: The __________ is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.
median
What is the relationship between the angles in a triangle and the sides according to the Law of Sines?
The ratio of the side lengths is equal to the ratio of the sines of their opposite angles.
True or False: The Law of Cosines can only be applied to triangles with one obtuse angle.
False
What is the formula to find the area of a triangle using base and height?
A = 1/2 * base * height.
Fill in the blank: In a triangle, the sum of the angles must always equal __________ degrees.
180
