91Ó°ÊÓ

The Leaning Tower of Pisa in Italy has a height of 183 feet and is \(4^{\circ}\) off vertical. Find the horizontal distance d that the top of the tower is off vertical.

MULTIPLE REPRESENTATIONSu are standing on a cliff above an ocean. You see a sailboat from your vantage point 30 feet above the ocean. a. Draw and label a diagram of the situation. b. Make a table showing the angle of depression and the length of your line of sight. Use the angles \(40^{\circ}\) , \(50^{\circ}, 60^{\circ}, 70^{\circ},\) and \(80^{\circ}\) . c. Graph the values you found in part (b), with the angle measures on the \(\mathrm{x}\) -axis. dredict the length of the line of sight when the angle of depression is \(30^{\circ}\) .

Use the Pythagorean Theorem (Theorem 9.1) to prove the Hypotenuse-Leg (HL) Congruence Theorem (Theorem 5.9)

You are fertilizing a triangular garden. One side of the garden is 62 \(\square\)feet long, and another side is 54 feet long. The angle opposite the 62 -foot side is \(58^{\circ} .\) a. Draw a diagram to represent this situation. b. Use the Law of Sines to solve the triangle from part (a). c. One bag of fertilizer covers an area of 200 square feet. How many bags of fertilizer will you need to cover the entire garden?

Your friend claims 72 and 75 cannot be part of a Pythagorean triple because \(72^{2} \square 75^{2}\) does not equal a positive integer squared. Is your friend correct? Explain your reasoning.

A golfer hits a drive 260 yards on a hole that is 400 yards long. The shot is \(15^{\circ}\) off target. a What is the distance x from the golfer's ball to the \(\square\)hole? b. Assume the golfer is able to hit the ball precisely the distance found in part (a). What is the maximum angle \(\theta\) (theta) by which the ball can be off target in order to land no more than 10 yards from the hole?

In Exercises 41 and \(42,\) use the given statements to prove the theorem. Given \(\triangle A B C\) is a right triangle. Altitude \(\overline{C D}\) is drawn to hypotenuse \(\overline{A B}\) . Prove the Geometric Mean (Altitude) Theorem (Theorem 9.7 by showing that \(\mathrm{CD}^{2}=\mathrm{AD} \cdot \mathrm{BD}\)

In Exercises 41 and \(42,\) use the given statements to prove the theorem. Given \(\triangle A B C\) is a right triangle. Altitude \(\overline{C D}\) is drawn to hypotenuse \(\overline{A B}\) . Prove the Geometric Mean (Leg) Theorem (Theorem 9.8 ) by showing that \(\mathrm{CB}^{2}=\mathrm{DB} \cdot \mathrm{AB}\) and \(\mathrm{AC}^{2}=\mathrm{AD} \cdot \mathrm{AB}\)

Draw a right isosceles triangle and label the two leg lengths x. Then draw the altitude to the hypotenuse and label its length y. Now, use the Right Triangle Similarity Theorem (Theorem 9.6) to draw the three similar triangles from the image and label any side length that is equal to either x or y. What can you conclude about the relationship between the two smaller triangles? Explain your reasoning.

The ambiguous case of \(\square\)the Law of Sines occurs when you are given the measure of one acute angle, the length of one adjacent side, and the length of the side opposite that angle, which is less than the length of the adjacent side. This results in two possible triangles. Using the given information, \(\square\)nd two possible solutions for \(\triangle \mathrm{ABC}\) Draw a diagram for each triangle. Draw a diagram for each triangle. (Hint. The inverse sine function gives only acute angle measures, so consider the acute angle and its supplement for \(\angle \mathrm{B}\) .) a. \(\mathrm{m} \angle \mathrm{A}=40^{\circ}, \mathrm{a}=13, \mathrm{b}=16\) b. \(\mathrm{m} \angle \mathrm{A}=21^{\circ}, \mathrm{a}=17, \quad \mathrm{b}=32\)