Question

application problems using similar triangles

1. if a tree casts a 24 - foot shadow at the same time that a yardstick casts a 2 - foot shadow, find the height of the tree.
2. a bush is sighted on the other side of a canyon. find the width of the canyon.
3. a 12 - centimeter rod is held between a flashlight and a wall as shown. find the length of the shadow on the wall if the rod is 45 cm from the wall and 15 cm from the light.
4. the cheerleaders at city high make their own megaphones by cutting off the small end of a cone made from heavy paper. if the small end of the megaphone is to have a radius of 2.5 cm, what should be the height of the cone that is cut off?
5. find the width of the brady river.
6. the foot of a ladder is 1.2 m from a fence that is 1.8 m high. the ladder touches the fence and rests against a building that is 1.8 m behind the fence. draw a diagram, and determine the height on the building reached by the top of the ladder.

Explanation:

Step1: Set up proportion for tree - height problem

Since the triangles formed by the tree and its shadow and the yardstick and its shadow are similar, the ratios of their corresponding sides are equal. A yardstick is 3 feet long. Let the height of the tree be $x$. The proportion is $\frac{3}{2}=\frac{x}{24}$.

Step2: Solve the proportion for $x$

Cross - multiply: $2x = 3\times24$. Then $2x=72$. Divide both sides by 2: $x = 36$ feet.

Step3: Set up proportion for canyon - width problem

Let the width of the canyon be $x$. The proportion from the similar triangles is $\frac{10}{7.5}=\frac{x}{100}$.

Step4: Solve the proportion for $x$

Cross - multiply: $7.5x=10\times100$. So $7.5x = 1000$. Then $x=\frac{1000}{7.5}=\frac{10000}{75}=\frac{400}{3}\approx133.33$ feet.

Step5: Set up proportion for shadow - length problem

Let the length of the shadow be $x$. The two similar triangles give the proportion $\frac{12}{x}=\frac{15}{15 + 45}$.

Step6: Solve the proportion for $x$

First simplify the right - hand side: $\frac{15}{15 + 45}=\frac{15}{60}=\frac{1}{4}$. Then cross - multiply: $x=12\times4 = 48$ cm.

Step7: Set up proportion for megaphone problem

Let the height of the cut - off cone be $h$. Using similar triangles of the whole cone and the cut - off part, we have $\frac{h}{60}=\frac{2.5}{56 + 2.5}=\frac{2.5}{58.5}=\frac{25}{585}=\frac{5}{117}$.

Step8: Solve the proportion for $h$

Cross - multiply: $117h=5\times60$. So $h=\frac{300}{117}=\frac{100}{39}\approx2.56$ cm.

Step9: Set up proportion for river - width problem

Let the width of the river be $x$. The similar triangles give the proportion $\frac{15}{x + 8}=\frac{8}{7}$.

Step10: Solve the proportion for $x$

Cross - multiply: $15\times7=8\times(x + 8)$. So $105=8x+64$. Subtract 64 from both sides: $8x=105 - 64 = 41$. Then $x=\frac{41}{8}=5.125$ m.

Step11: Set up proportion for ladder problem

Let the height on the building be $h$. The two similar right - angled triangles give the proportion $\frac{1.8}{h}=\frac{1.2}{1.2 + 1.8}$.

Step12: Solve the proportion for $h$

First simplify the right - hand side: $\frac{1.2}{1.2 + 1.8}=\frac{1.2}{3}=0.4$. Then cross - multiply: $0.4h=1.8$. Divide both sides by 0.4: $h = 4.5$ m.

Answer:

1. The height of the tree is 36 feet.
2. The width of the canyon is $\frac{400}{3}\approx133.33$ feet.
3. The length of the shadow is 48 cm.
4. The height of the cut - off cone is $\frac{100}{39}\approx2.56$ cm.
5. The width of the Brady River is 5.125 m.
6. The height on the building reached by the top of the ladder is 4.5 m.