High School Math : Triangles

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Triangles

Rt_triangle_lettersIn this figure, side \(\displaystyle X=20\)\(\displaystyle Y=13\), and \(\displaystyle c=90^\circ\). What is the value of angle \(\displaystyle a\)?

Possible Answers:

Undefined

\(\displaystyle 6.5^\circ\)

\(\displaystyle 90^\circ\)

\(\displaystyle 49.46^\circ\)

\(\displaystyle 40.54^\circ\)

Correct answer:

\(\displaystyle 49.46^\circ\)

Explanation:

Since \(\displaystyle c=90^\circ\), we know we are working with a right triangle.

That means that \(\displaystyle \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\).

In this problem, that would be:

\(\displaystyle \cos(a)=\frac{\text{Y}}{\text{X}}\)

Plug in our given values:

\(\displaystyle \cos(a)=\frac{\text{13}}{\text{20}}\)

\(\displaystyle \cos(a)=0.65\)

\(\displaystyle a=\cos^{-1}(0.65)\)

\(\displaystyle a=49.46^\circ\)

Example Question #1 : Triangles

Let ABC be a right triangle with sides \(\displaystyle A\) = 3 inches, \(\displaystyle B\) = 4 inches, and \(\displaystyle C\) = 5 inches. In degrees, what is the \(\displaystyle \sin \Theta\) where \(\displaystyle \Theta\) is the angle opposite of side \(\displaystyle A\)?

Possible Answers:

\(\displaystyle .7\)

\(\displaystyle .6\)

\(\displaystyle .4\)

\(\displaystyle .3\)

\(\displaystyle .5\)

Correct answer:

\(\displaystyle .6\)

Explanation:

3-4-5_triangle

We are looking for \(\displaystyle \sin(\theta )\). Remember the definition of \(\displaystyle sin\) in a right triangle is the length of the opposite side divided by the length of the hypotenuse. 

So therefore, without figuring out \(\displaystyle \Theta\) we can find

\(\displaystyle \sin \Theta =\frac{3}{5}=.6\)

Example Question #3 : Graphs And Inverses Of Trigonometric Functions

Rt_triangle_letters

In this figure, if angle \(\displaystyle c=90^\circ\), side \(\displaystyle X=12\), and side \(\displaystyle Z=8\), what is the measure of angle \(\displaystyle a\)?

Possible Answers:

\(\displaystyle 48.2^\circ\)

Undefined

\(\displaystyle \frac{2}{3}^\circ\)

\(\displaystyle 41.8^\circ\)

\(\displaystyle 90^\circ\)

Correct answer:

\(\displaystyle 41.8^\circ\)

Explanation:

Since \(\displaystyle c=90^\circ\), we know we are working with a right triangle.

That means that \(\displaystyle \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\).

In this problem, that would be:

\(\displaystyle \sin(a)=\frac{\text{Z}}{\text{X}}\)

Plug in our given values:

\(\displaystyle \sin(a)=\frac{8}{12}\)

\(\displaystyle \sin(a)=\frac{2}{3}\)

\(\displaystyle a=\sin^{-1}(\frac{2}{3})\)

\(\displaystyle a=41.8^\circ\)

Example Question #4 : Graphing The Sine And Cosine Functions

Rt_triangle_letters

In this figure, \(\displaystyle X=12\)\(\displaystyle Y=6\sqrt{3}\), and \(\displaystyle Z=6\). What is the value of angle \(\displaystyle a\)?

Possible Answers:

\(\displaystyle 45^\circ\)

\(\displaystyle 30^\circ\)

\(\displaystyle 60^\circ\)

\(\displaystyle 90^\circ\)

Undefined

Correct answer:

\(\displaystyle 30^\circ\)

Explanation:

Notice that these sides fit the pattern of a 30:60:90 right triangle: \(\displaystyle x:x\sqrt{3}:2x\).

In this case, \(\displaystyle x=6\).

Since angle \(\displaystyle a\) is opposite \(\displaystyle x\), it must be \(\displaystyle 30^\circ\).

Example Question #5 : Graphing The Sine And Cosine Functions

A triangle has angles of \(\displaystyle 30^\circ:60^\circ:90^\circ\). If the side opposite the \(\displaystyle 30^\circ\) angle is \(\displaystyle 7\), what is the length of the side opposite \(\displaystyle 60^\circ\)?

Possible Answers:

\(\displaystyle 7\sqrt{3}\)

\(\displaystyle 14\sqrt{3}\)

\(\displaystyle \frac{7\sqrt{2}}{2}\)

\(\displaystyle 14\)

\(\displaystyle 7\sqrt{2}\)

Correct answer:

\(\displaystyle 7\sqrt{3}\)

Explanation:

The pattern for \(\displaystyle 30^\circ:60^\circ:90^\circ\) is that the sides will be \(\displaystyle x:x\sqrt{3}:2x\).

If the side opposite \(\displaystyle 30^\circ\) is \(\displaystyle 7\), then the side opposite \(\displaystyle 60^\circ\) will be \(\displaystyle 7\sqrt{3}\).

Example Question #1 : Applying The Law Of Cosines

In \(\displaystyle \Delta ABC\)\(\displaystyle AB = 26\), \(\displaystyle BC = 32\), and \(\displaystyle AC = 23\). To the nearest tenth, what is \(\displaystyle m \angle A\) ?

Possible Answers:

\(\displaystyle 81.3 ^{\circ }\textrm{ or } 98.7^{ \circ }\)

\(\displaystyle 171.3^{ \circ }\)

\(\displaystyle 81.3 ^{\circ }\)

A triangle with these sidelengths cannot exist.

\(\displaystyle 98.7^{ \circ }\)

Correct answer:

\(\displaystyle 81.3 ^{\circ }\)

Explanation:

By the Triangle Inequality, this triangle can exist, since \(\displaystyle 23 + 26 > 32\).

By the Law of Cosines:

\(\displaystyle \left ( BC\right )^{2} = \left ( AB\right )^{2} + \left ( AC\right )^{2} - 2 \cdot AB \cdot AC \cdot \cos m \angle A\)

Substitute the sidelengths and solve for \(\displaystyle \cos m \angle A\) :

\(\displaystyle 32^{2} = 26^{2} + 23^{2} - 2\cdot 26 \cdot 23 \cdot \cos m \angle A\)

\(\displaystyle 1,024 =676 + 529 - 1,196 \cdot \cos m \angle A\)

\(\displaystyle 1,024 =1,205 - 1,196 \cdot \cos m \angle A\)

\(\displaystyle -181 = - 1,196 \cdot \cos m \angle A\)

\(\displaystyle \cos m \angle A = 181 \div 1,196 \approx 0.1513\)

\(\displaystyle m \angle A \approx \cos^{-1} 0.1513 \approx 81.3 ^{\circ }\)

Example Question #2 : Applying The Law Of Cosines

A triangle has sides of length 12, 17, and 22. Of the measures of the three interior angles, which is the greatest of the three?

Possible Answers:

\(\displaystyle 97.2^{\circ }\)

\(\displaystyle 86.4^{\circ }\)

\(\displaystyle 93.6^{\circ }\)

\(\displaystyle 119.2^{\circ }\)

\(\displaystyle 130.1^{\circ }\)

Correct answer:

\(\displaystyle 97.2^{\circ }\)

Explanation:

We can apply the Law of Cosines to find the measure of this angle, which we will call :

\(\displaystyle c^{2} = a^{2} + b^{2} -2ab \cos C\)

 

The widest angle will be opposite the side of length 22, so we will set:

\(\displaystyle c=22\)\(\displaystyle a=12\)\(\displaystyle b=17\)

 

\(\displaystyle 22^{2} = 12^{2} + 17^{2} -2\cdot 12\cdot 17 \cos C\)

\(\displaystyle 484 = 144 + 289 -408 \cos C\)

\(\displaystyle 51= -408 \cos C\)

\(\displaystyle \cos C = -\frac{51}{408} = -0.125\)

\(\displaystyle C = \cos^{-1} 0.125 = 97.2^{\circ }\)

 

Example Question #11 : Graphs And Inverses Of Trigonometric Functions

In \(\displaystyle \Delta ABC\)\(\displaystyle m \angle A = 66^{\circ }\) , \(\displaystyle AB = 26\), and \(\displaystyle AC = 23\). To the nearest tenth, what is \(\displaystyle BC\)?

Possible Answers:

\(\displaystyle 47.9\)

A triangle with these characteristics cannot exist.

\(\displaystyle 112.4\)

\(\displaystyle 26.8\)

\(\displaystyle 41.1\)

Correct answer:

\(\displaystyle 26.8\)

Explanation:

By the Law of Cosines:

\(\displaystyle \left ( BC\right )^{2} = \left ( AB\right )^{2} + \left ( AC\right )^{2} - 2 \cdot AB \cdot AC \cdot \cos m \angle A\)

or, equivalently,

\(\displaystyle BC =\sqrt{ \left ( AB\right )^{2} + \left ( AC\right )^{2} - 2 \cdot AB \cdot AC \cdot \cos m \angle A}\)

Substitute:

\(\displaystyle BC =\sqrt{ 26^{2} +23^{2} - 2 \cdot 26 \cdot 23 \cdot \cos 66 ^{\circ }}\)

\(\displaystyle \approx \sqrt{ 676 +529 - 1,196 \cdot 0.4067}\)

\(\displaystyle \approx \sqrt{ 718.6} \approx 26.8\)

Example Question #2 : Triangles

 

 

Rt_triangle_letters
In this figure, angle \(\displaystyle a=30^\circ\) and side \(\displaystyle Z=15\). If angle \(\displaystyle b=45^\circ\), what is the length of side \(\displaystyle Y\)?

Possible Answers:

\(\displaystyle 15\sqrt{2}\)

\(\displaystyle 30\)

\(\displaystyle \frac{15\sqrt{2}}{2}\)

\(\displaystyle 7.5\)

\(\displaystyle 30\sqrt{2}\)

Correct answer:

\(\displaystyle 15\sqrt{2}\)

Explanation:

For this problem, use the law of sines:

\(\displaystyle \frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }b}{\sin(b)}=\frac{\text{side opposite }c}{\sin(c)}\).

In this case, we have values that we can plug in:

\(\displaystyle \frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }b}{\sin(b)}\)

\(\displaystyle \frac{Z}{\sin(a)}=\frac{Y}{\sin{(b)}}\)

\(\displaystyle \frac{15}{\sin(30^\circ)}=\frac{Y}{\sin{(45^\circ)}}\)

\(\displaystyle \frac{15}{\frac{1}{2}}=\frac{Y}{\frac{\sqrt{2}}{2}}\)

Cross multiply:

\(\displaystyle 15*\frac{\sqrt{2}}{2}=\frac{1}{2}Y\)

Multiply both sides by \(\displaystyle 2\):

\(\displaystyle 15\sqrt{2}=Y\)

Example Question #3 : Triangles

Rt_triangle_letters

In this figure \(\displaystyle a=22^\circ\) and \(\displaystyle c=85^\circ\). If \(\displaystyle X=30\), what is \(\displaystyle Z\)?

Possible Answers:

\(\displaystyle 30.09\)

\(\displaystyle 11.28\)

\(\displaystyle 0.997\)

\(\displaystyle 20.1\)

\(\displaystyle 0.374\)

Correct answer:

\(\displaystyle 11.28\)

Explanation:

For this problem, use the law of sines:

\(\displaystyle \frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }b}{\sin(b)}=\frac{\text{side opposite }c}{\sin(c)}\).

In this case, we have values that we can plug in:

\(\displaystyle \frac{\text{side opposite }a}{\sin(a)}=\frac{\text{side opposite }c}{\sin(c)}\)

\(\displaystyle \frac{Z}{\sin(a)}=\frac{X}{\sin{(c)}}\)

\(\displaystyle \frac{Z}{\sin(22^\circ)}=\frac{30}{\sin{85^\circ}}\)

\(\displaystyle \frac{Z}{0.375}=\frac{30}{0.997}\)

\(\displaystyle \frac{Z}{0.375}=30.09\)

\(\displaystyle Z=30.09*0.375\)

\(\displaystyle Z=11.28\)

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