All Trigonometry Resources
Example Questions
Example Question #31 : Right Triangles
Which of the following is true about the right triangle below?
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse and the shortest side length is 2:1. Therefore, C = 2A.
Example Question #1 : Use Special Triangles To Make Deductions
Which of the following is true about the right triangle below?
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the shortest side length and the longer non-hypotenuse side length is . Therefore,
.
Example Question #2 : Use Special Triangles To Make Deductions
Which of the following is true about the right triangle below?
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse length and the second-longest side length is . Therefore,
.
Example Question #1 : Use Special Triangles To Make Deductions
Which of the following is true about the right triangle below?
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the two shorter side lengths are equal. Therefore, A = B.
Example Question #5 : Use Special Triangles To Make Deductions
Which of the following is true about the right triangle below?
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between a short side length and the hypotenuse is . Therefore,
.
Example Question #2 : Use Special Triangles To Make Deductions
Which of the following is true about the right triangle below?
The triangle is equilateral.
The triangle is isosceles.
The triangle is obtuse.
The triangle is scalene.
The triangle is isosceles.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between the two short side lengths is 1:1. Therefore, A = B. Triangles with two congruent side lengths are isosceles by definition.
Example Question #7 : Use Special Triangles To Make Deductions
In the figure below, is inscribed in a circle.
passes through the center of the circle. In
, the measure of
is twice the measure of
. The figure is drawn to scale.
Which of the following is true about the figure?
is equal in length to a radius of the circle.
is equal in length to a radius of the circle.
is equal in length to a diameter of the circle.
is equal in length to a diameter of the circle.
is equal in length to a radius of the circle.
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because is a diameter of the circle, arc
has a measure of 180 degrees. Therefore,
must be equal to
. Since
is a right triangle, the sum of its interior angles to 180 degrees. Since the measure of
is twice the measure of
,
. Therefore, the measure of
can be calculated as follows:
Therefore, is equal to
.
must be a 30-60-90 triangle. Therefore, side length
must be half the length of side length
, the hypotenuse of the triangle. Since
is a diameter of the circle, half of
represents the length of a radius of the circle. Therefore,
is equal in length to a radius of the circle.
Example Question #8 : Use Special Triangles To Make Deductions
In the figure below, is inscribed in a circle.
passes through the center of the circle. In
, the measure of
is twice the measure of
. The figure is drawn to scale.
Which of the following is true about the figure?
is isosceles.
is equilateral.
is a 45-45-90 triangle.
is a 30-60-90 triangle.
is a 30-60-90 triangle.
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because is a diameter of the circle, arc
has a measure of 180 degrees. Therefore,
must be equal to
. Since
is a right triangle, the sum of its interior angles equal 180 degrees. Since the measure of
is twice the measure of
,
. Therefore, the measure of
can be calculated as follows:
Therefore, is equal to
.
must be a 30-60-90 triangle.
Example Question #9 : Use Special Triangles To Make Deductions
In the figure below, is a diagonal of quadrilateral
.
has a length of 1.
and
are congruent and isosceles.
and
are perpendicular. The figure is drawn to scale.
Which of the following is a true statement?
is a 30-60-90 triangle.
and
, are parallel.
and
are perpendicular.
is equilateral.
and
, are parallel.
Since and
are perpendicular,
is a right angle. Since no triangle can have more than one right angle, and
is isosceles,
must be congruent to
. Since
is congruent to
and
measures 90 degrees,
and
can be calculated as follows:
Therefore, and
are both equal to 45 degrees.
is a 45-45-90 triangle. Since
is congruent to
,
is also a 45-45-90 triangle. The figure is drawn to scale, so
is a right angle. Since
has the same angle measure as
, the two angles are alternate interior angles and diagonal
is a transversal relative to
and
, which must be parallel.
Example Question #10 : Use Special Triangles To Make Deductions
In the figure below, is a diagonal of quadrilateral
.
has a length of
.
is congruent to
.
Which of the following is a true statement?
The perimeter of quadrilateral is
.
The area of quadrilateral is
.
The area of quadrilateral is
.
The perimeter of quadrilateral is
.
The area of quadrilateral is
.
Since and
are perpendicular,
is a right angle. Since no triangle can have more than one right angle, and
is isosceles,
must be congruent to
. Since angle CBD is congruent to
and
measures 90 degrees,
and
can be calculated as follows:
Therefore, and
are both equal to 45 degrees.
is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse
is
. Anyone of the four side lengths of quadrilateral
must, therefore, be equal to
. To find the area of
, multiply two side lengths:
.
Certified Tutor
All Trigonometry Resources
![Learning Tools by Varsity Tutors](https://vt-vtwa-app-assets.varsitytutors.com/assets/problems/og_image_practice_problems-9cd7cd1b01009043c4576617bc620d0d5f9d58294f59b6d6556fd8365f7440cf.jpg)