Triangles, being the focus of Trigonometry, deserve their own full section. Focus here is on types of triangles, the Pythagorean Theorem, the special triangles, and the area of a triangle (not requiring trigonometric functions). Many exercises involving the \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) and \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) that would normally be presented in Right Triangle Trigonometry are found in this section and its homework (but without reference to the trigonometric functions). Similar triangles are held off until next section.
To succeed in this section, you'll need to use some skills from previous courses. While you should already know them, this is the first time they've been required. You can review these skills in CRC's Corequisite Codex. If you have a support class, it might cover some, but not all, of these topics.
Prerequisite Skills and Support Topics (click to expand)
Solving Radical Equations: This skill is required when using the Pythagorean Theorem to find the length of a missing side of a right triangle, as the process often involves isolating a variable that is squared (e.g., Example 8).
Solving Compound Inequalities: This skill is necessary for applying the Triangle Inequality Theorem to find the possible range of lengths for a triangle's third side, as demonstrated in Example 3.
The following is a list of learning objectives for this section.
Learning Objectives (click to expand)
Find the missing angle within a triangle using the Triangle Sum.
Use the Triangle Inequality to determine constraints on an unknown side length of a triangle.
Use the Pythagorean Theorem to find the third side of a right triangle.
Find the missing sides of a \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) or \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle given one side.
Solve real-world applications requiring the Pythagorean Theorem or the special triangles.
The study of Trigonometry inevitably involves understanding everything we possibly can about triangles.
Definition: Triangle
When three line segments bound a portion of the plane, the resulting shape is called a triangle (also known as a three-sided polygon). The line segments are called the sides of the triangle. A point where two sides meet is called a vertex of the triangle. Finally, the angle formed in the triangle's interior where two sides meet is called an angle of the triangle.
In Figure \( \PageIndex{ 1 } \), the sides are denoted \( \overline{AB} \), \( \overline{BC} \), and \( \overline{AC} \). The vertices (plural of vertex) are \( A \), \( B \), and \( C \). The angles are denoted \( \angle A \), \( \angle B \), and \( \angle C \). Using the notation \( \triangle A B C \), we refer to the triangle by its vertices.
Figure \( \PageIndex{ 1 } \):\( \triangle A B C \)
If one tears off the corners of any triangle and lines them up, as shown in Figure \( \PageIndex{ 2 } \), they will always form a straight angle.
Figure \( \PageIndex{ 2 } \):Tearing corners from a triangle to form a straight angle
Theorem: Triangle Sum
The sum of the interior angles in a triangle is \( 180^{\circ} \).
Proof
Given the triangle, \( \triangle ABC \), draw a line parallel to side \( \overline{BC} \) going through vertex \( A \) (see the figure below).
Since \( \angle q + \angle a + \angle p \) forms a straight angle,\[ \angle q + \angle a + \angle p = 180^{\circ}. \nonumber \]Moreover, the line segment \( \overline{AC} \) is a transversal for the parallel lines \( \overline{BC} \) and \( \overline{DE} \). Therefore, \( \angle c = \angle p \). By a similar argument, \( \angle b = \angle q \). Hence,\[ \angle b + \angle a + \angle c = 180^{\circ}. \nonumber \]Thus, the sum of the angles of a triangle is \( 180^{\circ} \).
Example \(\PageIndex{1}\)
Two of the angles in the triangle shown below are \(25^{\circ}\) and \(115^{\circ}\). Find the third angle.
Figure \( \PageIndex{ 3 } \)
Solution
To find the third angle, we write an equation.\[\begin{array}{crrclr}
& & x+25+115 & = & 180 & (\text{Triangle Sum}) \\
\scriptscriptstyle\mathrm{Arithmetic} & \implies & x+140 & = & 180 & (\text{add}) \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \textrm{Algebra} & \implies & x & = & 40 & (\text{subtract }140\text{ from both sides})\\
\end{array} \nonumber \]The third angle is \(40^{\circ}\).
What does \( \mathrm{Arithmetic} \) and \( \xcancel{\textrm{Arithmetic}} \) mean?
To get you used to the Mathematical Mantra, every once in a while I will include the "thought processes" during solutions. Just to review, we perform Mathematics in the order we learned it - Arithmetic, Algebra, Trigonometry, \( \ldots \). At each step during a "mechanical" solution process, we should pause and ask ourselves if there is some simple Arithmetic to be done. If not, we move on to any Algebra that can be done. If there is no Algebra to be done, we then move on to Trigonometry.
Thus,
"\( \mathrm{Arithmetic} \implies\)" in a solution means that there is some Arithmetic we could do to clean up the previous expression
"\( \xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} \implies\)" means there isn't any Arithmetic but there is some Algebra we could use to clean up the previous expression or equation
"\( \xcancel{\mathrm{Arithmetic}} \to \xcancel{\mathrm{Algebra}} \to \mathrm{Trigonometry} \implies\)" means there isn't any Arithmetic nor is there any Algebra that can be done to clean up the previous step, so we need to start looking at doing something from Trigonometry.
In any triangle, the longest side is always opposite the largest angle, and the shortest side is opposite the smallest, as illustrated in Figure \( \PageIndex{ 6 } \). Although obvious to most people, this fact will become invaluable as we move through Trigonometry.
Figure \( \PageIndex{ 6 } \)
Notation: Angles and Sides of a Triangle
It is common (though not necessary) to label the angles of a triangle with capital letters and the side opposite each angle with the corresponding lowercase letter, as shown in the Figure \( \PageIndex{ 7 } \). We will follow this practice unless indicated otherwise.
Figure \( \PageIndex{ 7 } \)
Example \( \PageIndex{ 2 } \)
In \(\triangle F G H, \angle F=48^{\circ}\), and \(\angle G\) is obtuse. Side \(f\) is 6 feet long. What can be concluded about the other sides?
Solution
Because \(\angle G\) is greater than \(90^{\circ}\), we know that \(\angle F+\angle G\) is greater than \(90^{\circ}+48^{\circ}=138^{\circ}\), so \(\angle H\) is less than \(180^{\circ}-138^{\circ}=42^{\circ}\). Thus, \(\angle H<\angle F<\angle G\), and consequently \(h<f<g\). We can conclude that \(h<6\) feet long, and \(g>6\) feet long.
Checkpoint \( \PageIndex{ 2 } \)
In triangle \(\triangle R S T\), \(\angle S\) is \(72^{\circ}\). The remaining two angles have the same size. Which side is longer, \(s\) or \(t\)?
Answer
\(s\) is longer
The Triangle Inequality
The sum of the lengths of any two sides of a triangle must be greater than the third side; otherwise, the two sides will not meet to form a triangle. This fact is known as the triangle inequality. The theorem is presented without proof (however, you can always find a suitable proof online).
Theorem: Triangle Inequality
In any triangle, we must have that\[p + q > r, \nonumber \]where \(p, q\), and \(r\) are the lengths of the sides of the triangle (see the figure below).
That is, the sum of the lengths of any two sides of a triangle is greater than the length of the other side.
We cannot use the triangle inequality to find the exact lengths of a triangle's sides, but we can find the largest and smallest possible values for the length.
Example \( \PageIndex{ 3 } \)
Two sides of a triangle have lengths 7 inches and 10 inches, as shown below. What can be said about the length of the third side?
Figure \( \PageIndex{ 8 } \)
Solution
We let \(x\) represent the length of the third side of the triangle. By looking at each side in turn, we can apply the triangle inequality in three different ways to get\[7<x+10, \quad 10<x+7, \quad \text { and } \quad x<10+7 \nonumber\]We solve each of these inequalities to find\[-3<x, \quad 3<x, \quad \text { and } \quad x<17 \nonumber\]We already know that \(x>-3\) because \(x\) must be positive, but the other two inequalities do give us new information. The third side must be greater than 3 inches but less than 17 inches long.
Checkpoint \( \PageIndex{ 3 } \)
Can a triangle be made with three wooden sticks of lengths 14 feet, 26 feet, and 10 feet? Sketch a picture, and explain why or why not.
Answer
No, \(10+14\) is not greater than 26.
Types of Triangles
There are three types of triangles classified by whether they have three equal side lengths, two equal side lengths, or no side lengths that are equal. These are called the equilateral, isosceles, and scalene triangles, respectively.
Equilateral Triangles
Definition: Equilateral Triangle
A triangle in which all three sides have the same length is called an equilateral triangle.
Since all three sides of an equilateral triangle have the same length, there is no smallest side for which the smallest angle will be opposite, and there is no largest side for which the largest angle will be opposite. This gives us the following theorem.
Theorem: Angles of an Equilateral Triangle
All the angles of an equilateral triangle are equal.
Example \( \PageIndex{ 4 } \)
All three sides of a triangle are 4 feet long. Find the angles.
Figure \( \PageIndex{ 9 } \)
Solution
The triangle is equilateral, so all of its angles are equal. Thus\[\begin{array}{crrclr}
& & 3 x & = & 180 & (\text{Triangle Sum}) \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & x & = & 60 & (\text{divide both sides by }3) \\
\end{array} \nonumber \]Each of the angles is \(60^{\circ}\).
Checkpoint \(\PageIndex{4}\)
Find \(x, y\), and \(z\) in the triangle.
Figure \( \PageIndex{ 10 } \)
Answer
\(x=60^{\circ}, y=8, z=8\)
Isosceles Triangles
Definition: Isosceles Triangle
A triangle in which at least two sides are of equal length is called an isosceles triangle. The angle between the equal sides is called the vertex angle, and the other two angles are called the base angles.
Theorem: Angles of an Isosceles Triangle
The base angles of an isosceles triangle are equal.
In Example \( \PageIndex{ 5 } \), we introduce another common notation used throughout Geometry - the tick mark. We use tick marks on line segments to denote they have equal length. Similarly, tick marks on angles denote that the angles have the same measure.
Example \(\PageIndex{5}\)
Explain why \(\alpha=\beta\) in the triangle below.
Figure \( \PageIndex{ 11 } \)
Solution
Because they are the base angles of an isosceles triangle, \(\theta\) (theta) and \(\phi\) (phi) are equal. Also, \(\alpha=\theta\) because they are vertical angles, and similarly \(\beta=\phi\). Therefore, \(\alpha=\beta\) because they are equal to equal quantities.
Example \( \PageIndex{ 6 } \)
Find \(x\) and \(y\) in the triangle.
Figure \( \PageIndex{ 12 } \)
Solution
The triangle is isosceles, so the base angles are equal. Therefore, \(y=38^{\circ}\). To find the vertex angle, we solve\[ \begin{array}{crrclr}
& & x+38+38 & = & 180 & (\text{Triangle Sum}) \\
\scriptscriptstyle{\mathrm{Arithmetic}} & \implies & x+76 & = & 180 & (\text{add}) \\
\scriptscriptstyle{\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra}} & \implies & x & = & 104 & (\text{subtract }76\text{ from both sides}) \\
\end{array} \nonumber \]The vertex angle is \(104^{\circ}\).
Checkpoint \(\PageIndex{6}\)
Find \(x\) and \(y\).
Figure \( \PageIndex{ 13 } \)
Answer
\(x=140^{\circ}, y=9\)
Scalene Triangles
Finally, we have the scalene triangles.
Definition: Scalene Triangle
A triangle in which none of the sides have equal length (and, therefore, none of the angles are equal) is called a scalene triangle.
Figure \( \PageIndex{ 14 } \) shows a general scalene triangle. The different "hash marks" on the sides indicate the side lengths are different. The same is true for the angles.
Figure \( \PageIndex{ 14 } \):A general scalene triangle
Scalene triangles become important in this course when we learn the Law of Sines and the Law of Cosines.
Classification of Triangles by Largest Angle
Most people are not likely to refer to a triangle by its type (equilateral, isosceles, or scalene); however, knowing the features of each type of triangle is important in Trigonometry. Another classification system for triangles, and one that will be referenced constantly in Trigonometry, is based on the size of the triangle's largest angle. Three classifications exist: right, obtuse, and acute triangles.
Obtuse Triangles
We start with the class of triangles having their largest angle greater than \( 90^{ \circ } \).
Definition: Obtuse Triangle
A triangle which contains an obtuse angle is called an obtuse triangle.
Recall that an obtuse angle has measure strictly between \( 90^{ \circ } \) and \( 180^{ \circ } \) (meaning that we exclude \( 90^{ \circ } \) and \( 180^{ \circ } \)). Since the sum of the interior angles of a triangle must be \( 180^{ \circ } \), and one of the angles in an obtuse triangle is greater than \( 90^{ \circ } \), the two smaller angles in an obtuse triangle must sum to strictly less than \( 90^{ \circ } \).
Theorem: Angles of an Obtuse Triangle
In an obtuse triangle, one angle must be greater than \( 90^{ \circ } \), and the remaining two angles must each be less than \( 90^{ \circ } \).
Right Triangles
Reducing the largest possible angle to exactly \( 90^{ \circ } \), we arrive at arguably the most essential triangle classification in Trigonometry - the right triangles.
Definition: Right Triangle
A triangle containing a right angle is called a right triangle.
Example \( \PageIndex{ 7 } \)
One of the smaller angles of a right triangle is \(34^{\circ}\). What is the third angle?
Figure \( \PageIndex{ 15 } \)
Solution
The sum of the two smaller angles in a right triangle is \(90^{\circ}\). So\[\begin{array}{crrclr}
& & x+34 & = & 90 & \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & x & = & 56 & (\text{subtract }34\text{ from both sides}) \\
\end{array} \nonumber \]The unknown angle must be \(56^{\circ}\).
Checkpoint \(\PageIndex{7}\)
Two angles of a triangle are \(35^{\circ}\) and \(45^{\circ}\). Can it be a right triangle?
Answer
No.
Right Triangles: The Pythagorean Theorem
Arguably, the most important theorem in Trigonometry is the Pythagorean Theorem, mainly because without it, Trigonometry would not exist.
Theorem: Pythagorean Theorem
In a right triangle, if \(c\) is the length of the hypotenuse, and the lengths of the two legs are denoted by \(a\) and \(b\) (as shown in the figure below), then\[a^2 + b^2 = c^2. \nonumber \]
The proof of the Pythagorean Theorem is left as a guided exercise in the homework.
Example \( \PageIndex{ 8 } \)
A 25-foot ladder is placed against a wall so that its foot is 7 feet from the base of the wall. How far up the wall does the ladder reach?
Solution
We sketch the situation, as shown below, and label any known dimensions. We will call the unknown height \(h\).
Figure \( \PageIndex{ 16 } \)
The ladder forms the hypotenuse of a right triangle, so we can apply the Pythagorean Theorem. Substituting 25 for \(c, 7\) for \(b\), and \(h\) for \(a\), we get the following:\[ \begin{array}{rrrclcl}
& & a^2+b^2 & = & c^2 & & \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & h^2+7^2 & = & 25^2 & \quad & \left( \text{substitution} \right)\\
\scriptscriptstyle\mathrm{Arithmetic} & \implies & h^2+49 & = & 625 & \quad & \left( \text{apply powers} \right) \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & h^2 & = & 576 & \quad & \left(\text{subtract }49\text{ from both sides}\right) \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & h & = & \pm \sqrt{576} & \quad & \left(\text{Extraction of Roots}\right) \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & h & = & \pm 24 & \quad & \left(\text{simplify the radical}\right) \\
\end{array} \nonumber \]The height must be a positive number, so the solution \(-24\) does not make sense for this problem. The ladder reaches 24 feet up the wall.
Checkpoint \( \PageIndex{ 8 } \)
A baseball diamond is a square with sides that are 90 feet long. When the catcher sees a runner on first trying to steal second, he throws the ball to the second baseman. Find the straight-line distance from home plate to second base.
Answer
\(90 \sqrt{2} \approx 127.3\) feet
Caution
The Pythagorean Theorem is valid only for right triangles!
Most people do not realize that the Pythagorean Theorem is a bijection. That is, the converse of the theorem is also true. If the sides of a triangle satisfy the relationship \(a^2+b^2=c^2\), then the triangle must be a right triangle with hypotenuse \( c \). We can use this fact to test whether a given triangle has a right angle.
Example \( \PageIndex{ 9 } \)
Mike is paving a patio in his backyard and would like to know if the corner at \(C\) is a right angle. He measures 20 cm along one side from the corner and 48 cm along the other, placing pegs \(P\) and \(Q\) at each position, as shown below. The line joining those two pegs is 52 cm long. Is the corner a right angle?
Figure \( \PageIndex{ 17 } \)
Solution
If it is a right triangle, its sides must satisfy \(p^2+q^2=c^2\). We find\[\begin{array}{rcl}
p^2+q^2 & = & 20^2+48^2 \\
& = & 400+2304 \\
& = & 2704 \\
\end{array} \nonumber \]and\[\begin{array}{rcl}
c^2 & = & 52^2 \\
& = & 2704 \\
\end{array} \nonumber \]Therefore, yes, because \(p^2+q^2=c^2\), the corner at \(C\) is a right angle.
Checkpoint \( \PageIndex{ 9 } \)
The sides of a triangle measure 15, 25, and 30 inches long. Is the triangle a right triangle?
Answer
No
The Pythagorean Theorem relates the sides of right triangles. However, for information about the sides of other triangles, the best we can do (without Trigonometry) is the triangle inequality. Moreover, the Pythagorean Theorem doesn't help us find the angles in a triangle - that comes later.
Acute Triangles
Finally, we reduce the largest possible angle within a triangle to strictly less than \( 90^{ \circ } \) and obtain the acute triangles.
Definition: Acute Triangle
A triangle in which all angles are less than \( 90^{ \circ } \) is called an acute triangle.
Special Triangles
We finally arrive at what could be considered the most essential triangles in elementary Trigonometry - the special triangles. These triangles are revered in Trigonometry because they allow the learner (read as "you") to perform computations without needing a calculator. This, in turn, allows you to focus on concepts rather than computations.
In order to derive one of the theorems we are about to introduce, we need a quick definition.
Definition: Altitude of a Triangle
The altitude of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side (or the line containing the opposite side).
The altitude of a triangle introduces a right angle, as the following theorem states.
Theorem: Altitudes and Right Angles
The altitude of a triangle represents the shortest distance from the vertex to the opposite side and forms a \( 90^{ \circ } \) angle with that side.
A right triangle in which one angle is \( 30^{ \circ } \) and another angle is \( 60^{ \circ } \) is called a \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) triangle.
Working with this triangle allows us to know the length of all three sides if only given the length of one side.
Theorem: Side Relationships for a \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) Triangle
In any right triangle in which the two acute angles are \( 30^{ \circ } \) and \( 60^{ \circ } \), the hypotenuse is always twice the length of the shortest side (the side opposite the \( 30^{ \circ } \) angle), and the remaining side (opposite the \( 60^{ \circ } \) angle) is always \( \sqrt{3} \) times the shortest side.
Proof
Consider an equilateral triangle with side lengths \( 2a \), as shown in the figure below.
Since this triangle is equilateral, each angle is \( 60^{ \circ } \). Drawing an altitude divides the triangle into two smaller triangles. Since the altitude is perpendicular to the base, each of these smaller triangles is a right triangle. Moreover, since one angle in each of these smaller right triangles is \( 60^{ \circ } \), the remaining angle must be \( 30^{ \circ } \). Furthermore, the side opposite the \( 30^{ \circ } \) angle must have length \( a \) (in the original equilateral triangle, the side opposite \( 60^{ \circ } \) had length \( 2a \), so it should make sense that the side opposite \( 30^{ \circ } \) is half of this). We now consider one of these smaller right triangles (see the figure below).
The missing side length, which we will call \( x \) for now, can be found using the Pythagorean Theorem.\[\begin{array}{rrcl}
& x^2 + a^2 & = & (2a)^2 \\
\implies & x^2 + a^2 & = & 4a^2 \\
\implies & x^2 & = & 3a^2 \\
\implies & x & = & a \sqrt{3} \\
\end{array} \nonumber \]
Before we go any further, notice in the proof of this theorem the convention of writing \( x = a \sqrt{3} \) instead of \( x = \sqrt{3} a \). This is purposeful. The latter often leads the reader to think that the \( a \) is "under" the square root. It's good form always to write the radical at the end of a term to avoid this visual ambiguity.
Example \( \PageIndex{ 10 } \)
Matterhorn Chocolate bars are sold in boxes shaped like triangular prisms. The two triangular ends of a box are equilateral triangles with 8-centimeter sides. When stacking the boxes on a shelf, the second level will be a triangle’s altitude above the bottom level. Find the exact length of the triangle’s altitude, \(h\).
Solution
The altitude divides the triangle into two \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangles, as shown in Figure \( \PageIndex{ 18 } \) below.
Figure \( \PageIndex{ 18 } \)
We see that the hypotenuse is \( 2a = 8 \), which implies the shortest side (opposite the \( 30^{ \circ } \) angle) is \( a=4 \). Hence, the side opposite the \( 60^{ \circ } \) angle is \( a\sqrt{3} = 4 \sqrt{3} \approx 6.9282 \) centimeters long.
Checkpoint \( \PageIndex{ 10 } \)
A ladder is leaning against a house so that the top is 12 feet above the ground, and the bottom makes a \( 60^{ \circ } \) angle with the ground. Assuming the wall of the house is perpendicular to the ground (and the ground is level), how long is the ladder? State your answer in exact form and round to two decimal places.
Answer
\(\dfrac{24}{\sqrt{3}} \approx 13.86 \) feet
Note that the exact form of the answer in Checkpoint \( \PageIndex{ 10 } \) was left as \( \frac{24}{\sqrt{3}} \). While it is true that you could rationalize the denominator to get\[\dfrac{24}{\sqrt{3}} = \dfrac{24}{\sqrt{3}} \cdot \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{24 \sqrt{3}}{3} = 8\sqrt{3}, \nonumber \]we will commonly leave radicals in the denominators of purely numeric expressions.
The second of our our two special triangles is the \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle. This is also known as an isosceles right triangle.
Definition: Isosceles Right Triangle
An isosceles triangle in which the vertex angle is \( 90^{ \circ } \) is called an isosceles right triangle.
Since the right angle takes \( 90^{ \circ } \), the remaining two angles must share the remaining \( 90^{ \circ } \) evenly. That is, the remaining two angles must be \( 45^{ \circ } \) each.
Since an isosceles right triangle has two angles that are \( 45^{ \circ } \), it is commonly called a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle.
Just as with our first special triangle, knowing one side of a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle allows us to easily compute the lengths of the remaining sides.
Theorem: Side Relationship for a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) Triangle
In any right triangle in which the two acute angles are \( 45^{ \circ } \), the hypotenuse is always \( \sqrt{2} \) times the side length.
Proof
The sides opposite the \( 45^{ \circ } \) angles in a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle must have the same length. Let \( a \) be the lengths of these sides and \( x \) be the length of the hypotenuse, as shown in the figure below.
Since this is a right triangle, we can use the Pythagorean Theorem to find the length of the hypotenuse in terms of \( a \).\[\begin{array}{rrcl}
& a^2 + a^2 & = & x^2 \\
\implies & 2 a^2 & = & x^2 \\
\implies & a \sqrt{2} & = & x \\
\end{array} \nonumber \]
Example \(\PageIndex{11}\)
You are hired to build a truss to support a balcony. Figure \( \PageIndex{ 19 } \) shows the design plans you have submitted for the project. What is the length of the beam needed from point \( A \) to point \( D \)?
Figure \( \PageIndex{ 19 } \)
Solution
\( \triangle ABD \) is a \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle. Therefore, if the leg length (opposite the \( 45^{ \circ } \) angle) is \( a = 4 \) meters, then beam between points \( A \) and \( D \) must measure \( a\sqrt{2} = 4 \sqrt{2} \approx 5.66 \) meters.
Checkpoint \(\PageIndex{11}\)
Suppose, instead of knowing the 4-meter lengths in Example \( \PageIndex{ 11 } \), we knew that the length of the beam from \( A \) to \( D \) was 12 meters. How long is the beam from \( B \) to \( D \)?
Answer
\( \dfrac{12}{\sqrt{2}} \approx 8.49 \) meters
Area of a Triangle
Our final review topic for triangles in this section is something necessary at multiple points throughout Trigonometry and Calculus - the area of a triangle.
Theorem: Area of a Triangle
Let \( b \) be the length of one side of a triangle and \( h \) be the altitude height from that side to its opposite vertex. Then the area of the triangle is given by\[ A = \dfrac{1}{2} bh. \nonumber \]
Proof
To be done in homework.
Example \(\PageIndex{12}\)
Compute the area of the triangle in Figure \( \PageIndex{20} \).
Figure \( \PageIndex{ 20 } \)
Solution
We are given the length of one leg of this triangle as 11 inches. Let's call this leg the base. Thus, \( b = 11\). The altitude from the base to its opposite vertex has length 8 inches. Hence, \( h = 8 \). Therefore, the area of this triangle is\[ A = \dfrac{1}{2} b h = \dfrac{1}{2} (11)(8) = 44 \, \text{square inches.} \nonumber \]
Checkpoint \(\PageIndex{12}\)
Compute the area of the triangle in Figure \( \PageIndex{21} \) if we know that \( c = 5.24 \) and \(h = 1.25\).
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