How To Find The Shortest Side Of A Triangle

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Subsection Introduction

From geometry nosotros know that the sum of the angles in a triangle is 180°. Are there any relationships between the angles of a triangle and its sides?

Showtime of all, you have probably observed that the longest side in a triangle is always reverse the largest angle, and the shortest side is contrary the smallest angle, equally illustrated beneath.

Example ii.two .

In \(\triangle FGH, \angle F=48\degree,\) and \(\bending Thousand\) is obtuse. Side \(f\) is 6 feet long. What tin can you conclude about the other sides?

Solution.

Considering \(\bending G\) is greater than \(ninety\degree\text{,}\) nosotros know that \(\angle F +\bending One thousand\) is greater than \(90\caste + 48\degree = 138\degree\text{,}\) so \(\angle F\) is less than \(180\degree-138\degree = 42\degree.\) Thus, \(\angle H \lt \angle F \lt \angle G,\) and consequently \(h \lt f \lt g\text{.}\) We can conclude that \(h \lt 6\) feet long, and \(g \gt vi\) feet long.

Checkpoint 2.3 .

In isosceles triangle \(\triangle RST\text{,}\) the vertex angle \(\angle S = 72\degree\text{.}\) Which side is longer, \(s\) or \(t\text{?}\)

Subsection The Triangle Inequality

Information technology is also truthful that the sum of the lengths of any 2 sides of a triangle must exist greater than the third side, or else the two sides will non meet to class a triangle. This fact is called the triangle inequality.

Triangle Inequality.

In any triangle, we must have that

\begin{equation*} p+q \gt r \end{equation*}

where \(p, q, \text{and}~ r\) are the lengths of the sides of the triangle.

We cannot use the triangle inequality to find the exact lengths of the sides of a triangle, but we can find largest and smallest possible values for the length.

Example 2.four .

Two sides of a triangle have lengths 7 inches and 10 inches, as shown at right. What can you say about the length of the 3rd side?

Solution.

We let \(x\) represent the length of the 3rd side of the triangle. Past looking at each side in turn, nosotros tin apply the triangle inequality three different ways, to get

\begin{equation*} 7 \lt x+10, ~~~ ten \lt ten+7, ~~~ \text{and} ~~~ 10 \lt 10+7 \stop{equation*}

We solve each of these inequalities to observe

\begin{equation*} -three \lt x, ~~~ iii \lt x, ~~~ \text{and} ~~~ ten \lt 17 \stop{equation*}

We already know that \(10 \gt -3\) because \(x\) must be positive, but the other ii inequalities do give us new data. The third side must be greater than iii inches but less than 17 inches long.

Checkpoint ii.five .

Can you lot make a triangle with three wooden sticks of lengths 14 feet, 26 feet, and 10 feet? Sketch a picture, and explain why or why not.

Answer.

No, \(10+fourteen\) is non greater than 26.

Subsection Correct Triangles: The Pythagorean Theorem

In Affiliate 1 we used the Pythagorean theorem to derive the distance formula. Nosotros tin also apply the Pythagorean theorem to notice one side of a correct triangle if we know the other two sides.

Pythagorean Theorem.

In a right triangle, if \(c\) stands for the length of the hypotenuse, and the lengths of the two legs are denoted by \(a\) and \(b\text{,}\) and then

\brainstorm{equation*} \blert{a^two + b^two = c^two} \terminate{equation*}

Instance ii.6 .

A 25-human foot ladder is placed against a wall so that its pes is vii feet from the base of the wall. How far up the wall does the ladder reach?

Solution.

We brand a sketch of the state of affairs, as shown below, and label whatever known dimensions. We'll phone call the unknown meridian \(h\text{.}\)

The ladder forms the hypotenuse of a right triangle, and so we can apply the Pythagorean theorem, substituting 25 for \(c\text{,}\) seven for \(b\text{,}\) and \(h\) for \(a\text{.}\)

\begin{align*} a^2 + b^2 \amp = c^ii\\ h^ii + 7^ii \amp = 25^ii \end{align*}

Now solve by extraction of roots:

\begin{marshal*} h^2 + 49 \amp = 625 \amp\amp \blert{\text{Decrease 49 from both sides.}}\\ h^two \amp = 576 \amp\amp \blert{\text{Extract roots.}}\\ h \amp = \pm \sqrt{576} \amp\amp \blert{\text{Simplify the radical.}}\\ h \amp = \pm 24 \terminate{align*}

The acme must be a positive number, and then the solution \(-24\) does not make sense for this trouble. The ladder reaches 24 feet up the wall.

Checkpoint 2.vii .

A baseball diamond is a square whose sides are 90 anxiety long. The catcher at abode plate sees a runner on outset trying to steal second base of operations, and throws the brawl to the second-baseman. Discover the straight-line distance from home plate to 2d base of operations.

Respond.

\(90\sqrt{two} \approx 127.3\) feet

Case 2.9 .

Delbert is paving a patio in his dorsum thou, and would like to know if the corner at \(C\) is a correct angle.

He measures 20 cm forth ane side from the corner, and 48 cm along the other side, placing pegs \(P\) and \(Q\) at each position, as shown at right. The line joining those 2 pegs is 52 cm long. Is the corner a right angle?

Solution.

If is a right triangle, then its sides must satisfy \(p^2 + q^two = c^two\text{.}\) We observe

\begin{align*} p^2 + q^two \amp = 20^2 + 48^2 = 400 + 2304 = 2704\\ c^ii \amp = 52^ii = 2704 \terminate{align*}

Yes, because \(p^2 + q^ii = c^ii\text{,}\) the corner at \(C\) is a correct angle.

Checkpoint 2.10 .

The sides of a triangle measure out 15 inches, 25 inches, and 30 inches long. Is the triangle a right triangle?

The Pythagorean theorem relates the sides of right triangles. However, for information nearly the sides of other triangles, the all-time nosotros can do (without trigonometry!) is the triangle inequality. Nor does the Pythagorean theorem assistance us notice the angles in a triangle. In the side by side section nosotros discover relationships betwixt the angles and the sides of a right triangle.

Review the following skills you will need for this section.

Algebra Refresher 2.2 .

1. \(\:6-x \gt 3\)

2. \(\:\dfrac{-3x}{4} \ge -six\)

three. \(\:3x-7 \le -ten\)

4. \(\:4-3x \lt 2x+9\)

If \(x \lt 0\text{,}\) which of the following expressions are positive, and which are negative?

5. \(\:-10\)

half dozen. \(\:-(-x)\)

7. \(\:\abs{x}\)

eight. \(\:-\abs{ten}\)

9. \(\:-\abs{-x}\)

10. \(\:x^{-i}\)

\(\underline{\qquad\qquad\qquad\qquad}\)

Algebra Refresher Answers

1. \(\displaystyle \:10 \lt 3\)

2. \(\displaystyle \:10 \le 8\)

3. \(\displaystyle \:ten \le -ane\)

4. \(\displaystyle \:x \gt -1\)

5. Positive

6. Negative

7. Positive

8. Negative

9. Negative

10. Negative

Subsection Section ii.1 Summary

Subsubsection Vocabulary

• Converse

• Extraction of roots

• Inequality

Subsubsection Concepts

1. The longest side in a triangle is opposite the largest angle, and the shortest side is opposite the smallest angle.

2. Triangle Inequality: In any triangle, the sum of the lengths of whatsoever two sides is greater than the length of the third side.

3. Pythagorean Theorem: In a right triangle with hypotenuse \(c,~~ a^ii +b^2 = c^2\text{.}\)

4. If the sides of a triangle satisfy the human relationship \(~a^2 +b^two = c^ii~\text{,}\) then the triangle is a correct triangle.

Subsubsection Study Questions

1. Is information technology ever true that the hypotenuse is the longest side in a right triangle? Why or why not?

2. In \(\triangle DEF\text{,}\) is it possible that \(~d+e\gt f~\) and \(~e+f\gt d~\) are both true? Explain your answer.

3. In a correct triangle with hypotenuse \(c\text{,}\) we know that \(~a^2 +b^2 = c^ii~\text{.}\) Is information technology also truthful that \(~a + b = c~\text{?}\) Why or why not?

4. The two shorter sides of an obtuse triangle are iii in and four in. What are the possible lengths for the third side?

Subsubsection Skills

1. Identify inconsistencies in figures #1-12

2. Use the triangle inequality to put bounds on the lengths of sides #13-16

3. Apply the Pythagorean theorem to observe the sides of a right triangle #17-26

4. Use the Pythagorean theorem to identify correct triangles #27-32

5. Solve problems using the Pythagorean theorem #33-42

Exercises Homework 2.i

Exercise Group.

For Problems 1–12, explain why the measurements shown cannot exist accurate.

1.

2.

3.

4.

5.

6.

7.

8.

nine.

ten.

eleven.

12.

13.

If 2 sides of a triangle are 6 feet and ten anxiety long, what are the largest and smallest possible values for the length of the tertiary side?

14.

Ii adjacent sides of a parallelogram are 3 cm and 4 cm long. What are the largest and smallest possible values for the length of the diagonal?

15.

If one of the equal sides of an isosceles triangle is 8 millimeters long, what are the largest and smallest possible values for the length of the base?

16.

The town of Madison is 15 miles from Newton, and twenty miles from Lewis. What are the possible values for the distance from Lewis to Newton?

Practice Group.

For Problems 17–22,

1. Make a sketch of the situation described, and label a correct triangle.

2. Employ the Pythagorean Theorem to solve each problem.

17.

The size of a TV screen is the length of its diagonal. If the width of a 35-inch TV screen is 28 inches, what is its summit?

18.

If a thirty-meter pine tree casts a shadow of 30 meters, how far is the tip of the shadow from the elevation of the tree?

19.

The diagonal of a square is 12 inches long. How long is the side of the foursquare?

xx.

The length of a rectangle is twice its width, and its diagonal is meters long. Notice the dimensions of the rectangle.

21.

What size rectangle can be inscribed in a circle of radius 30 feet if the length of the rectangle must be three times its width?

22.

What size square tin can be inscribed inside a circle of radius eight inches, so that its vertices just touch the circumvolve?

Exercise Group.

For Bug 23–26, observe the unknown side of the triangle.

Exercise Group.

For Problems 27–32, decide whether a triangle with the given sides is a right triangle.

27.

9 in, 16 in, 25 in

28.

12 m, sixteen chiliad, twenty m

29.

5 thou, 12 grand, xiii m

thirty.

5 ft, 8 ft, thirteen ft

31.

\(5^ii\) ft, \(8^two\) ft, \(13^ii\) ft

32.

\(\sqrt{5}\) ft, \(\sqrt{8}\) ft, \(\sqrt{13}\) ft

33.

Show that the triangle with vertices \((0,0)\text{,}\) \((6,0)\) and \((iii,3)\) is an isosceles right triangle, that is, a correct triangle with two sides of the same length.

34.

Two opposite vertices of a foursquare are \(A(-9,-5)\) and \(C(3,3)\text{.}\)

1. Detect the length of a diagonal of the square.

2. Discover the length of the side of the square.

35.

A 24-foot flagpole is being raised by a rope and pulley, as shown in the figure. The loose finish of the rope can exist secured to a ring on the ground vii feet from the base of the pole. From the ring to the elevation of the pulley, how long should the rope be when the flagpole is vertical?

36.

To check whether the corners of a frame are square, carpenters sometimes measure out the sides of a triangle, with two sides coming together at the join of the boards. Is the corner shown in the figure square?

Practise Group.

37.

Observe \(\alpha, \beta\) and \(h\text{.}\)

38.

Observe \(\blastoff, \beta\) and \(d\text{.}\)

39.

Find the diagonal of a cube of side 8 inches. Hint: Find the diagonal of the base get-go.

40.

Notice the diagonal of a rectangular box whose sides are 6 cm by 8 cm by x cm. Hint: Find the diagonal of the base of operations commencement.

Practice Group.

For Problems 41 and 42, make a sketch and solve.

41.

1. The back of Brian's pickup truck is 5 anxiety broad and seven anxiety long. He wants to bring home a 9-foot length of copper pipe. Volition it lie flat on the floor of the truck?

2. Discover the length of the side of the foursquare.

42.

What is the longest curtain rod that volition fit inside a box lx inches long past x inches wide by 4 inches tall?

43.

In this problem, we'll show that any angle inscribed in a semi-circumvolve must be a right angle. The effigy shows a triangle inscribed in a unit circle, one side lying on the diameter of the circumvolve and the reverse vertex at point \((p,q)\) on the circle.

1. What are the coordinates of the other two vertices of the triangle? What is the length of the side joining those vertices?

2. Employ the distance formula to compute the lengths of the other two sides of the triangle.

3. Evidence that the sides of the triangle satisfy the Pythagorean theorem, \(a^2 + b^two = c^2\text{.}\)

44.

There are many proofs of the Pythagorean theorem. Here is a elementary visual statement.

1. What is the length of the side of the large square in the figure? Write an expression for its area.

2. Write another expression for the area of the large square past adding the areas of the four right triangles and the smaller central square.

3. Equate your two expressions for the area of the large foursquare, and deduce the Pythagorean theorem.

Source: https://yoshiwarabooks.org/trig/Side-and-Angle-Relationships.html

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