Triangles

1

While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. You are standing feet from the base of the platform, and the angle of elevation from your position to the top of the platform is degrees. How many feet tall is the platform?

Explanation

You can draw the following right triangle using the information given by the question:

Since you want to find the height of the platform, you will need to use tangent.

$\tan 62=\frac{x}{200}$

Make sure to round to places after the decimal.

2

You need to build a diagonal support for the bleachers at the local sportsfield. The support needs to reach from the ground to the top of the bleacher. How the support should look is highlighted in blue below. The bleacher wall is 10 feet high and perpendicular to the ground. The owner would like the support to only stick out 3 feet from the bleacher at the bottom. What is the length of the support you need to build?

Screen shot 2020 08 27 at 1.35.40 pm

20 ft

10.44 ft

109 ft

11.32 ft

Explanation

It is important to recognize that the bleacher, the ground, and the support form a right triangle with the right angle formed by the intersection of the bleacher wall and the ground. We know the bottom of the support should only be 3ft from the bleacher wall on the ground and the bleacher wall is 10ft high. We will use the Pythagorean Theorem to solve for the length of the support, which is the hypotenuse of this right triangle. Our base of the triangle is 3 feet and the leg is 10 feet.

$a^{2}+b^{2}=c^{2}$

$3^{2}+10^{2}=c^{2}$

$109=c^{2}$

$\sqrt{109}=\sqrt{c^{2}}$

And so we need a support of 10.44 feet long.

3

While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. You are standing feet from the base of the platform, and the angle of elevation from your position to the top of the platform is degrees. How many feet tall is the platform?

Explanation

You can draw the following right triangle using the information given by the question:

Since you want to find the height of the platform, you will need to use tangent.

$\tan 62=\frac{x}{200}$

Make sure to round to places after the decimal.

4

You need to build a diagonal support for the bleachers at the local sportsfield. The support needs to reach from the ground to the top of the bleacher. How the support should look is highlighted in blue below. The bleacher wall is 10 feet high and perpendicular to the ground. The owner would like the support to only stick out 3 feet from the bleacher at the bottom. What is the length of the support you need to build?

Screen shot 2020 08 27 at 1.35.40 pm

20 ft

10.44 ft

109 ft

11.32 ft

Explanation

It is important to recognize that the bleacher, the ground, and the support form a right triangle with the right angle formed by the intersection of the bleacher wall and the ground. We know the bottom of the support should only be 3ft from the bleacher wall on the ground and the bleacher wall is 10ft high. We will use the Pythagorean Theorem to solve for the length of the support, which is the hypotenuse of this right triangle. Our base of the triangle is 3 feet and the leg is 10 feet.

$a^{2}+b^{2}=c^{2}$

$3^{2}+10^{2}=c^{2}$

$109=c^{2}$

$\sqrt{109}=\sqrt{c^{2}}$

And so we need a support of 10.44 feet long.

5

You need to build a diagonal support for the bleachers at the local sportsfield. The support needs to reach from the ground to the top of the bleacher. How the support should look is highlighted in blue below. The bleacher wall is 10 feet high and perpendicular to the ground. The owner would like the support to only stick out 3 feet from the bleacher at the bottom. What is the length of the support you need to build?

Screen shot 2020 08 27 at 1.35.40 pm

20 ft

10.44 ft

109 ft

11.32 ft

Explanation

It is important to recognize that the bleacher, the ground, and the support form a right triangle with the right angle formed by the intersection of the bleacher wall and the ground. We know the bottom of the support should only be 3ft from the bleacher wall on the ground and the bleacher wall is 10ft high. We will use the Pythagorean Theorem to solve for the length of the support, which is the hypotenuse of this right triangle. Our base of the triangle is 3 feet and the leg is 10 feet.

$a^{2}+b^{2}=c^{2}$

$3^{2}+10^{2}=c^{2}$

$109=c^{2}$

$\sqrt{109}=\sqrt{c^{2}}$

And so we need a support of 10.44 feet long.

6

If c=70, a=50, and $\angle A=122^\circ$ find $\angle C$ to the nearest degree.

$8.11^\circ$

$49.89^\circ$

$8.11^\circ$ and $171.89^\circ$

$49.89^\circ$ and $130.11^\circ$

no solution

Explanation

Notice that the given information is Angle-Side-Side, which is the ambiguous case. Therefore, we should test to see if there are no triangles that satisfy, one triangle that satisfies, or two triangles that satisfy this. From $\frac{\sin A}{a}=\frac{\sin C}{c}$ , we get $c\cdot \frac{\sin A}{a}=\sin C$ . In this equation, if $\sin C >1$ , no $\angle A$ that satisfies the triangle can be found. If $\sin C=1, C=90^\circ$ and there is a right triangle determined. Finally, if $\sin C <1$ , two measures of $\angle C$ can be calculated: an acute $\angle C$ and an obtuse angle $C'=180^\circ-C$ . In this case, there may be one or two triangles determined. If $C'+A \geq 180^\circ$ , then the $\angle C'$ is not a solution.

In this problem, $\sin C = 1.18>1$ , which means that there are no solutions to $\angle C$ that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem.

7

While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. You are standing feet from the base of the platform, and the angle of elevation from your position to the top of the platform is degrees. How many feet tall is the platform?

Explanation

You can draw the following right triangle using the information given by the question:

Since you want to find the height of the platform, you will need to use tangent.

$\tan 62=\frac{x}{200}$

Make sure to round to places after the decimal.

8

If c=70, a=50, and $\angle A=122^\circ$ find $\angle C$ to the nearest degree.

$8.11^\circ$

$49.89^\circ$

$8.11^\circ$ and $171.89^\circ$

$49.89^\circ$ and $130.11^\circ$

no solution

Explanation

Notice that the given information is Angle-Side-Side, which is the ambiguous case. Therefore, we should test to see if there are no triangles that satisfy, one triangle that satisfies, or two triangles that satisfy this. From $\frac{\sin A}{a}=\frac{\sin C}{c}$ , we get $c\cdot \frac{\sin A}{a}=\sin C$ . In this equation, if $\sin C >1$ , no $\angle A$ that satisfies the triangle can be found. If $\sin C=1, C=90^\circ$ and there is a right triangle determined. Finally, if $\sin C <1$ , two measures of $\angle C$ can be calculated: an acute $\angle C$ and an obtuse angle $C'=180^\circ-C$ . In this case, there may be one or two triangles determined. If $C'+A \geq 180^\circ$ , then the $\angle C'$ is not a solution.

In this problem, $\sin C = 1.18>1$ , which means that there are no solutions to $\angle C$ that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem.

9

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

Screen shot 2020 08 27 at 4.39.20 pm

Which of the following is a true statement?

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

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: $\frac{1}{\sqrt{2}}*\frac{1}{\sqrt{2}}=\frac{1}{2}$ .

10

If c=70, a=50, and $\angle A=122^\circ$ find $\angle C$ to the nearest degree.

$8.11^\circ$

$49.89^\circ$

$8.11^\circ$ and $171.89^\circ$

$49.89^\circ$ and $130.11^\circ$

no solution

Explanation

Notice that the given information is Angle-Side-Side, which is the ambiguous case. Therefore, we should test to see if there are no triangles that satisfy, one triangle that satisfies, or two triangles that satisfy this. From $\frac{\sin A}{a}=\frac{\sin C}{c}$ , we get $c\cdot \frac{\sin A}{a}=\sin C$ . In this equation, if $\sin C >1$ , no $\angle A$ that satisfies the triangle can be found. If $\sin C=1, C=90^\circ$ and there is a right triangle determined. Finally, if $\sin C <1$ , two measures of $\angle C$ can be calculated: an acute $\angle C$ and an obtuse angle $C'=180^\circ-C$ . In this case, there may be one or two triangles determined. If $C'+A \geq 180^\circ$ , then the $\angle C'$ is not a solution.

In this problem, $\sin C = 1.18>1$ , which means that there are no solutions to $\angle C$ that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem.

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation