Basic Geometry : Triangles

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #1 : 45/45/90 Right Isosceles Triangles

What is the length of the diagonal of a square with sides of \displaystyle 8\; in?

Possible Answers:

\displaystyle 8\sqrt{2}\; in

\displaystyle 4\sqrt{2}\; in

\displaystyle 8\sqrt{3}\; in

\displaystyle 4\sqrt{3}\; in

\displaystyle 12\; in

Correct answer:

\displaystyle 8\sqrt{2}\; in

Explanation:

The diagonal cuts a square into two \displaystyle 45-45-90 right triangles with the legs of the triangle being the sides of the square and the hypotenuse is the diagonal of the triangle.

Using Pythagorean Theorem we get:

\displaystyle x^{2}=8^{2}+ 8^{2} or \displaystyle x=8\sqrt{2}

Example Question #1 : Triangles

Img050

\displaystyle What\;is\;the\;length\;of\;side\;\overline{AC}?

Possible Answers:

\displaystyle 3\sqrt3

\displaystyle 2

\displaystyle 3\sqrt2

\displaystyle 3

Correct answer:

\displaystyle 3\sqrt2

Explanation:

\displaystyle Pythagorean\;Theorem: A^2+B^2=C^2

\displaystyle 3^2+3^2=C^2

\displaystyle 9+9=C^2

\displaystyle 18=C^2

\displaystyle C=\sqrt18= \sqrt{9\times2} = \sqrt9\times\sqrt2= 3\sqrt2

Example Question #2 : 45/45/90 Right Isosceles Triangles

A right triangle has two sides of length \displaystyle s; what is the correct formula for finding the length of the hypotenuse \displaystyle h?

Possible Answers:

\displaystyle h=2s^2

\displaystyle h=\sqrt{2s}

\displaystyle h=\sqrt{\frac{s^2}{2}}}

\displaystyle h=\sqrt{2s^2}

Correct answer:

\displaystyle h=\sqrt{2s^2}

Explanation:

The Pythagorean Theorem states that the sum of the squares of the sides of the triangle are equal to the square of the hypotenuse, summed up as \displaystyle a^2+b^2=c^2 but for the case of right isoceles triangles where a and b are equal, it can be rewritten as:

 \displaystyle s^2+s^2=h^2,

which simplifies to 

\displaystyle h^2=2s^2.

\displaystyle h=\sqrt2s^2

Example Question #3 : 45/45/90 Right Isosceles Triangles

\displaystyle \Delta ABC is a \displaystyle 45^\circ-45^\circ-90^\circ triangle.

\displaystyle m\angle B=90^\circ

\displaystyle \overline{AB}=3

Triangles_5

Using the Pythagorean Theorem, calculate the length of the hypotenuse of \displaystyle \Delta ABC.

Possible Answers:

\displaystyle \sqrt{2}

\displaystyle 3\sqrt{2}

There is not enough information given to answer this question.

\displaystyle \sqrt{6}

Correct answer:

\displaystyle 3\sqrt{2}

Explanation:

The Pythagorean Theorem holds that, in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.  That is,

\displaystyle a^2+b^2=c^2

where \displaystyle c is the hypotenuse and \displaystyle a and \displaystyle b are the other two sides.

Here, \displaystyle c corresponds to \displaystyle \overline{AC}, and, since this is a \displaystyle 45^{\circ}-45^{\circ}-90^{\circ} triangle, the length of \displaystyle a equals the length of \displaystyle b, so we know that \displaystyle \overline{AB}=\overline{BC}=3.

Apply the Pythagorean Theorem.

\displaystyle \left |AB \right |^2+\left |BC \right |^2=\left |AC \right |^2

\displaystyle 3^2+3^2=\left | AC\right |^2

\displaystyle 9+9=\left | AC\right |^2

\displaystyle 18=\left | AC\right |^2

\displaystyle \sqrt{18}=AC

\displaystyle \sqrt{9\cdot2}=AC

\displaystyle 3\sqrt{2}=AC

So the length of the hypotenuse is \displaystyle 3\sqrt{2}.

Example Question #4 : 45/45/90 Right Isosceles Triangles

The following image is not to scale.

Find the length of the hypotenuse of the right triangle:

 

Find_the_hypotenuse

Possible Answers:

\displaystyle 14

\displaystyle 12

\displaystyle \pm13.9

\displaystyle 13.7

\displaystyle 13.9

Correct answer:

\displaystyle 13.9

Explanation:

Because the problem states that this the triangle is a right triangle and provides the length of the two legs, the third side (hypotenuse) can be calculated using the Pythagorean Theorem. 

\displaystyle a^{2}+b^{2}=c^{2}, where a and b can be arbitrarily chosen for either leg length and c represents the length of the hypotenuse. 

\displaystyle 5^{2}+13^{2}=c^{2}

\displaystyle 194=c^{2}

\displaystyle \sqrt{194}=\sqrt{c^{2}}

\displaystyle \pm{\color{Magenta} 13.9}=c but because the question requires to find the length of the hypotenuse, c cannot be a negative number. Therefore the final answer is +13.9.

Example Question #2 : Triangles

The following image is not to scale.

Find the length of the hypotenuse of the \displaystyle 45/45/90 right triangle. 

Find_the_hyp

Possible Answers:

\displaystyle 2\sqrt2in

\displaystyle 2.9 in

\displaystyle -2\sqrt2in

\displaystyle 2\sqrt3in

\displaystyle 2.75 in

Correct answer:

\displaystyle 2\sqrt2in

Explanation:

Find_the_hyp

45/45/90 triangles are always isosceles. This means that two of the legs of the triangle are congruent. In the figure, it's indicates which two sides are congruent.

From here, we can find the length of the hypotenuse through the Pythagorean Theorem. We can confirm this because the problem has given us no angle measures to perform trig functions with.

\displaystyle a^2+b^2=c^2

\displaystyle 2^2+2^2=c^2

\displaystyle 4+4=c^2

\displaystyle 8=c^2

\displaystyle \sqrt{8}=\sqrt{c^2}

\displaystyle {\color{Blue} 2\sqrt{2}in}=c 

c also equals -2√2, but because the hypotenuse is a physical length, the value cannot be negative distance. 

Example Question #1 : Triangles

Square \displaystyle ABCD has a side length of \displaystyle 3. What is the length of its diagonal?

Possible Answers:

\displaystyle 3\sqrt{2}

\displaystyle 3\sqrt{3}

Cannot be determined from the information provided

\displaystyle 5

\displaystyle 4

Correct answer:

\displaystyle 3\sqrt{2}

Explanation:

The answer can be found two different ways. The first step is to realize that this is really a triangle question, even though it starts with a square. By drawing the square out and adding the diagonal, you can see that you form two right triangles. Furthermore, the diagonal bisects two ninety-degree angles, thereby making the resulting triangles a \displaystyle 45-45-90 triangle. 

From here you can go one of two ways: using the Pythagorean Theorem to find the diagonal, or recognizing the triangle as a \displaystyle 45-45-90 triangle.

1) Using the Pythagorean Theorem

Once you recognize the right triangle in this question, you can begin to use the Pythagorean Theorem. Remember the formula: \displaystyle a^{2}+b^{2}=c^{2}}, where \displaystyle a and \displaystyle b are the lengths of the legs of the triangle, and \displaystyle c is the length of the triangle's hypotenuse.

In this case, \displaystyle a=b=3. We can substitute these values into the equation and then solve for \displaystyle c, the hypotenuse of the triangle and the diagonal of the square:

\displaystyle 3^{2}+3^{2}=c^{2}}

\displaystyle 18=c^2

\displaystyle c=\sqrt{18}

\displaystyle c=\sqrt{9*2}

\displaystyle c=\sqrt{9}*{\sqrt{2}}

\displaystyle c=3*\sqrt{2}

\displaystyle c=3\sqrt2

The length of the diagonal is \displaystyle 3\sqrt2.

 

2) Using Properties of \displaystyle 45-45-90 Triangles

The second approach relies on recognizing a \displaystyle 45-45-90 triangle. Although one could solve this rather easily with Pythagorean Theorem, the following method could be faster.

\displaystyle 45-45-90 triangles have side length ratios of \displaystyle x:x:x\sqrt2, where \displaystyle x represents the side lengths of the triangle's legs and \displaystyle x\sqrt2 represents the length of the hypotenuse.

In this case, \displaystyle x=3 because it is the side length of our square and the triangles formed by the square's diagonal.

Therefore, using the \displaystyle 45-45-90 triangle ratios, we have \displaystyle 3\sqrt{2} for the hypotenuse of our triangle, which is also the diagonal of our square.

Example Question #1 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

What is the length of the hypotenuse of an isosceles right triangle with an area of \displaystyle 10.125\:in?

Possible Answers:

\displaystyle 18\sqrt{3}\:in

\displaystyle 9.25\sqrt{2}\:in

\displaystyle 9\:in

\displaystyle 4.5\sqrt{2}\:in

\displaystyle 4.5\sqrt{5}\:in

Correct answer:

\displaystyle 4.5\sqrt{2}\:in

Explanation:


Recall that an isosceles right triangle is also a \displaystyle 45-45-90 triangle. It has sides that appear as follows:

_tri51

Therefore, the area of the triangle is:

\displaystyle A=\frac{1}{2}x^2, since the base and the height are the same.

For our data, this means:

\displaystyle 10.125=\frac{1}{2}x^2

Solving for \displaystyle x, you get:

\displaystyle 20.25=x^2

\displaystyle x=4.5

So, your triangle looks like this:

_tri31

Now, you can solve this with a ratio and easily find that it is \displaystyle 4\sqrt{2}.  You also can use the Pythagorean Theorem. To do the latter, it is:

\displaystyle h^2 = 4.5^2+4.5^2

\displaystyle h = \sqrt{4.5^2+4.5^2}

Now, just do your math carefully:

\displaystyle h = \sqrt{4.5^2(1+1)}

That is a weird kind of factoring, but it makes sense if you distribute back into the group. This means you can simplify:

\displaystyle h = \sqrt{4.5^2(2)} = \sqrt{4.5^2}\sqrt{2}=4.5\sqrt{2}\:in 

Example Question #4 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

When the sun shines on a \displaystyle 4ft pole, it leaves a shadow on the ground that is also \displaystyle 4ft. What is the distance from the top of the pole to the end of its shadow?

Possible Answers:

\displaystyle 4 ft

\displaystyle 16 ft

\displaystyle 5.22 ft

\displaystyle 5.66 ft

\displaystyle 6.14 ft

Correct answer:

\displaystyle 5.66 ft

Explanation:

The pole and its shadow make a right angle. Because they are the same length, they form an isosceles right triangle (45/45/90). We can use the Pythagorean Theorem to find the hypotenuse. \displaystyle a^2+b^2=c^2. In this case, \displaystyle a=b=4. Therefore, we do \displaystyle 4^2+4^2=c^2=32. So \displaystyle c=\sqrt{32}\approx5.66

Example Question #2 : Triangles

Find the length of the hypotenuse.

1

Possible Answers:

\displaystyle 5

\displaystyle 3\sqrt{3}

\displaystyle \sqrt{10}

\displaystyle 5\sqrt2

Correct answer:

\displaystyle 5\sqrt2

Explanation:

To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of \displaystyle a and \displaystyle b and a hypotenuse of \displaystyle c,

13

\displaystyle c^2=a^2+b^2

\displaystyle c=\sqrt{a^2+b^2}

Now, plug in the values of \displaystyle a and \displaystyle b from the question.

\displaystyle \text{Hypotenuse}=\sqrt{5^2+5^2}

\displaystyle \text{Hypotenuse}=\sqrt{25+25}

Solve.

\displaystyle \text{Hypotenuse}=\sqrt{50}

Simplify.

\displaystyle \text{Hypotenuse}=5\sqrt2

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