# Trigonometry : Use Special Triangles To Make Deductions

## Example Questions

### Example Question #40 : Right Triangles

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

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?

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?

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?

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?

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?

The triangle is equilateral.

The triangle is isosceles.

The triangle is scalene.

The triangle is obtuse.

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.

Which of the following is true about the figure?

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.

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.

Which of the following is true about the figure?

is isosceles.

is a 30-60-90 triangle.

is equilateral.

is a 45-45-90 triangle.

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.

Which of the following is a true statement?

is a 30-60-90 triangle.

and , are parallel.

is equilateral.

and  are perpendicular.

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 .

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 .