Trigonometry : Use Special Triangles To Make Deductions

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #40 : Right Triangles

Which of the following is true about the right triangle below?

Screen shot 2020 08 27 at 10.51.31 am

Possible Answers:

Correct answer:

Explanation:

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?

Screen shot 2020 08 27 at 4.25.30 pm

Possible Answers:

Correct answer:

Explanation:

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?

Screen shot 2020 08 27 at 3.45.47 pm

Possible Answers:

Correct answer:

Explanation:

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 #3 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

Screen shot 2020 08 27 at 10.57.48 am

Possible Answers:

Correct answer:

Explanation:

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 #4 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

Possible Answers:

Correct answer:

Explanation:

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 #5 : Use Special Triangles To Make Deductions

Which of the following is true about the right triangle below?

Screen shot 2020 08 27 at 2.29.27 pm

Possible Answers:

The triangle is equilateral.

The triangle is isosceles.

The triangle is scalene.

The triangle is obtuse.

Correct answer:

The triangle is isosceles.

Explanation:

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 #6 : 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.

Screen shot 2020 08 27 at 11.23.21 am

Which of the following is true about the figure?

Possible Answers:

 is equal in length to a diameter 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 radius of the circle.

Correct answer:

 is equal in length to a radius of the circle.

Explanation:

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 #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.

Screen shot 2020 08 27 at 2.01.34 pm

Which of the following is true about the figure?

Possible Answers:

 is isosceles.

 is a 30-60-90 triangle.

 is equilateral.

 is a 45-45-90 triangle.

Correct answer:

 is a 30-60-90 triangle.

Explanation:

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 #8 : 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.

Screen shot 2020 08 28 at 9.51.38 am

Which of the following is a true statement?

Possible Answers:

 is a 30-60-90 triangle.

 and , are parallel.

 is equilateral.

 and  are perpendicular.

Correct answer:

 and , are parallel.

Explanation:

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 #1 : Use Special Triangles To Make Deductions

In the figure below,  is a diagonal of quadrilateral .  has a length of is congruent to .

Screen shot 2020 08 27 at 4.39.20 pm

Which of the following is a true statement?

Possible Answers:

The perimeter of quadrilateral  is .

The area of quadrilateral  is .

The area of quadrilateral  is .

The perimeter of quadrilateral  is .

Correct answer:

The area of quadrilateral  is .

Explanation:

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: .

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