From geometry we 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? First of all, you have probably observed that the longest side in a triangle is always opposite the largest angle, and the shortest side is opposite the smallest angle, as illustrated below. In \(\triangle FGH, \angle F=48\degree,\) and \(\angle G\) is obtuse. Side \(f\) is 6 feet long. What can you conclude about the other sides? Solution. Because \(\angle G\) is greater than \(90\degree\text{,}\) we know that \(\angle F +\angle G\) is greater than \(90\degree + 48\degree = 138\degree\text{,}\) 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 \(g \gt 6\) feet long. In isosceles triangle \(\triangle RST\text{,}\) the vertex angle \(\angle S = 72\degree\text{.}\) Which side is longer, \(s\) or \(t\text{?}\) It is also true that the sum of the lengths of any two sides of a triangle must be greater than the third side, or else the two sides will not meet to form a triangle. This fact is called the triangle inequality. 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. 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 three different ways, to get \begin{equation*} 7 \lt x+10, ~~~ 10 \lt x+7, ~~~ \text{and} ~~~ x \lt 10+7 \end{equation*} We solve each of these inequalities to find \begin{equation*} -3 \lt x, ~~~ 3 \lt x, ~~~ \text{and} ~~~ x \lt 17 \end{equation*} We already know that \(x \gt -3\) 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. Can you 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, \(10+14\) is not greater than 26. In Chapter 1 we used the Pythagorean theorem to derive the distance formula. We can also use the Pythagorean theorem to find one side of a right triangle if we know the other two sides. 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 make a sketch of the situation, as shown below, and label any known dimensions. We'll call the unknown height \(h\text{.}\) \begin{align*} h^2 + 49 \amp = 625 \amp\amp \blert{\text{Subtract 49 from both sides.}}\\ h^2 \amp = 576 \amp\amp \blert{\text{Extract roots.}}\\ h \amp = \pm \sqrt{576} \amp\amp \blert{\text{Simplify the radical.}}\\ h \amp = \pm 24 \end{align*} 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. A baseball diamond is a square whose sides are 90 feet long. The catcher at home plate sees a runner on first trying to steal second base, and 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 Keep in mind that the Pythagorean theorem is true only for right triangles, so the converse of the theorem is also true. In other words, if the sides of a triangle satisfy the relationship \(a^2 + b^2 = c^2\text{,}\) then the triangle must be a right triangle. We can use this fact to test whether or not a given triangle has a right angle. Delbert is paving a patio in his back yard, and would like to know if the corner at \(C\) is a right angle. Solution.
If is a right triangle, then its sides must satisfy \(p^2 + q^2 = c^2\text{.}\) We find \begin{align*} p^2 + q^2 \amp = 20^2 + 48^2 = 400 + 2304 = 2704\\ c^2 \amp = 52^2 = 2704 \end{align*} Yes, because \(p^2 + q^2 = c^2\text{,}\) the corner at \(C\) is a right angle. The sides of a triangle measure 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 about the sides of other triangles, the best we can do (without trigonometry!) is the triangle inequality. Nor does the Pythagorean theorem help us find the angles in a triangle. In the next section we discover relationships between the angles and the sides of a right triangle. Review the following skills you will need for this section.
For Problems 1–12, explain why the measurements shown cannot be accurate.
If two sides of a triangle are 6 feet and 10 feet long, what are the largest and smallest possible values for the length of the third side? Two 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? 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? The town of Madison is 15 miles from Newton, and 20 miles from Lewis. What are the possible values for the distance from Lewis to Newton?
For Problems 17–22,
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 height? If a 30-meter pine tree casts a shadow of 30 meters, how far is the tip of the shadow from the top of the tree? The diagonal of a square is 12 inches long. How long is the side of the square? The length of a rectangle is twice its width, and its diagonal is meters long. Find the dimensions of the rectangle.
For Problems 23–26, find the unknown side of the triangle.
For Problems 27–32, decide whether a triangle with the given sides is a right triangle.
9 in, 16 in, 25 in 12 m, 16 m, 20 m 5 m, 12 m, 13 m 5 ft, 8 ft, 13 ft \(5^2\) ft, \(8^2\) ft, \(13^2\) ft \(\sqrt{5}\) ft, \(\sqrt{8}\) ft, \(\sqrt{13}\) ft Show that the triangle with vertices \((0,0)\text{,}\) \((6,0)\) and \((3,3)\) is an isosceles right triangle, that is, a right triangle with two sides of the same length. Two opposite vertices of a square are \(A(-9,-5)\) and \(C(3,3)\text{.}\)
Find \(\alpha, \beta\) and \(h\text{.}\) Find \(\alpha, \beta\) and \(d\text{.}\)
For Problems 41 and 42, make a sketch and solve.
The back of Brian's pickup truck is five feet wide and seven feet long. He wants to bring home a 9-foot length of copper pipe. Will it lie flat on the floor of the truck? Find the length of the side of the square. What is the longest curtain rod that will fit inside a box 60 inches long by 10 inches wide by 4 inches tall?
There are many proofs of the Pythagorean theorem. Here is a simple visual argument.
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Triangle ABC is a right triangle what is the relationship between angles A and B
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