Use Special Triangles To Make Deductions

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Trigonometry › Use Special Triangles To Make Deductions

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In the figure below, $\overline{BD}$ is a diagonal of quadrilateral . $\overline{BD}$ has a length of . $\angle CBD$ is congruent to $\angle BDC$ .

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Which of the following is a true statement?

The area of quadrilateral is $\frac{1}{2}$ .

CORRECT

The area of quadrilateral is .

The perimeter of quadrilateral is .

Explanation

Since and are perpendicular, $\angle BCD$ is a right angle. Since no triangle can have more than one right angle, and $\Delta BCD$ is isosceles, $\angle CBD$ must be congruent to $\angle BDC$ . Since angle CBD is congruent to $\angle BDC$ and $\angle BCD$ measures 90 degrees, $\angle CBD$ and $\angle BDC$ can be calculated as follows:

Therefore, and are both equal to 45 degrees. $\Delta BCD$ is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse is $1:\sqrt{2}$ . Anyone of the four side lengths of quadrilateral must, therefore, be equal to $\frac{BD}{\sqrt{2}}=\frac{1}{\sqrt{2}}$ . To find the area of , multiply two side lengths: $\frac{1}{\sqrt{2}}*\frac{1}{\sqrt{2}}=\frac{1}{2}$ .

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

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CORRECT

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.

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

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$\sqrt{3} * A = 2B$

$\sqrt{3} * B = 2C$

$\2A = \sqrt{3} * B$

$\2B = \sqrt{3} * C$

CORRECT

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 $2:\sqrt{3}$ . Therefore, $2B=\sqrt{3}*C$ .

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

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The triangle is scalene.

The triangle is isosceles.

CORRECT

The triangle is equilateral.

The triangle is obtuse.

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.

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

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$A = (\frac{1}{2})B$

CORRECT

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.

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

$C=\sqrt{2}*B$

CORRECT

$B=\sqrt{2}*A$

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 $1:\sqrt{2}$ . Therefore, $C=\sqrt{2}*A$ .

In the figure below, $\Delta ABC$ is inscribed in a circle. $\overline{AC}$ passes through the center of the circle. In $\Delta ABC$ , the measure of $\angle C$ is twice the measure of $\angle A$ . The figure is drawn to scale.

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Which of the following is true about the figure?

$\Delta ABC$ is isosceles.

$\Delta ABC$ is equilateral.

$\Delta ABC$ is a 30-60-90 triangle.

CORRECT

$\Delta ABC$ is a 45-45-90 triangle.

Explanation

For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because $\overline{AC}$ is a diameter of the circle, arc has a measure of 180 degrees. Therefore, $\angle ABC$ must be equal to $180*\frac{1}2{}=90^{\circ}$ . Since $\Delta ABC$ is a right triangle, the sum of its interior angles equal 180 degrees. Since the measure of $\angle C$ is twice the measure of $\angle A$ , . Therefore, the measure of $\angle A$ can be calculated as follows:

Therefore, $\angle C$ is equal to $2*30=60^{\circ}$ . $\Delta ABC$ must be a 30-60-90 triangle.

Screen shot 2020 08 27 at 11.23.21 am

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.

CORRECT

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, $\angle ABC$ must be equal to $180*\frac{1}2{}=90^{\circ}$ . Since $\Delta ABC$ is a right triangle, the sum of its interior angles to 180 degrees. Since the measure of $\angle C$ is twice the measure of $\angle A$ , . Therefore, the measure of $\angle A$ can be calculated as follows:

Therefore, $\angle C$ is equal to $2*30=60^{\circ}$ . $\Delta ABC$ must be a 30-60-90 triangle. Therefore, side length $\overline{BC}$ must be half the length of side length $\overline{AC}$ , the hypotenuse of the triangle. Since $\overline{AC}$ is a diameter of the circle, half of $\overline{AC}$ represents the length of a radius of the circle. Therefore, $\overline{BC}$ is equal in length to a radius of the circle.

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

Screen shot 2020 08 27 at 4.25.30 pm

$A = \sqrt{3} * B$

$B = \sqrt{3} * A$

CORRECT

$C = \sqrt{3}* B$

$B = \sqrt{3} * C$

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 $1:\sqrt{3}$ . Therefore, $B=\sqrt{3}*A$ .

In the figure below, $\overline{BD}$ is a diagonal of quadrilateral . $\overline{BD}$ has a length of 1. $\Delta ABD$ and $\Delta BCD$ are congruent and isosceles. $\overline{BC}$ and $\overline{CD}$ are perpendicular. The figure is drawn to scale.

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Which of the following is a true statement?

$\overline{AD}$ and $\overline{BC}$ , are parallel.

CORRECT

$\overline{AB}$ and $\overline{BD}$ are perpendicular.

$\Delta ABD$ is a 30-60-90 triangle.

$\Delta BCD$ is equilateral.

Explanation

Since $\overline{BC}$ and $\overline{CD}$ are perpendicular, $\angle BCD$ is a right angle. Since no triangle can have more than one right angle, and $\Delta BCD$ is isosceles, $\angle CBD$ must be congruent to $\angle BDC$ . Since $\angle CBD$ is congruent to $\angle BDC$ and $\angle BCD$ measures 90 degrees, $\angle CBD$ and $\angle BDC$ can be calculated as follows:

Therefore, $\angle CBD$ and $\angle BDC$ are both equal to 45 degrees. $\Delta BCD$ is a 45-45-90 triangle. Since $\Delta ABD$ is congruent to $\Delta BCD$ , $\Delta ABD$ is also a 45-45-90 triangle. The figure is drawn to scale, so $\angle DAB$ is a right angle. Since $\angle ADB$ has the same angle measure as $\angle DBC$ , the two angles are alternate interior angles and diagonal $\overline{BD}$ is a transversal relative to $\overline{AD}$ and $\overline{BC}$ , which must be parallel.