Similar triangles word problems worksheet with answers

Problem 1 :

A girl is in the beach with her father. She spots a swimmer drowning. She shouts to her father who is 50 m due west of her. Her father is 10 m nearer to a boat than the girl. If her father uses the boat to reach the swimmer, he has to travel a distance 126 m from the boat. At the same time, the girl spots a man riding a water craft who is 98 m away from the boat. The man n the water craft is due east of the swimmer. How far must the man travel to rescue the swimmer?

Solution :

Let “A” be the place where her father is standing

Let “C” be the place where a girl is standing

Let “B” be the place where the boat is

Let D” be the place where water craft is

Let “E” be the place where the swimmer is

BC = x m

AB = (x - 10) m

By considering he triangles ∆ ABC, ∆ DBE

∠ABC = ∠DBE (vertically opposite angles)

∠BAC = ∠BDE (alternate angles)

By using AA similarity criterion ∆ ABC ~ ∆ DBE

(AB/DB) = (BC/BE) = (AC/DE)

(AB/DB) = (BC/BE)

(x – 10)/98 = x/126

126 (x – 10) = 98 x

126 x – 1260 = 98 x

126 x – 98 x = 1260

28 x = 1260

x = 1260/28

x = 45

BC = 45 m

Also, (BC/BE) = (AC/DE)

DE = (AC x BE)/BC

= (50 x 126)/45

= 6300/45

= 140 m

The man has to travel 140 m to rescue the swimmer.

Problem 2 :

P and Q are points on sides AB and AC respectively, of triangle ABC. If AP = 3 cm, PB = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ.

Solution :

AP/PB = AQ/AC

3/9 = 5/15

1/3 = 1/3

In triangles APQ, and ABC we get

(AP/AB) = (AQ/AC)

∠A = ∠A

By using SAS criterion ∆ APQ ~ ∆ ABC

(AP/AB) = (Q/AC) = (PQ/BC)

(AP/PB) = (PQ/BC)

PQ/BC = 3/9

PQ/BC = 1/3

3PQ = BC

BC = 3PQ

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Problem 1 :

A ramp is built enable wheel-chair access to a building that is 24 cm above ground level. The ramp has a constant slope of 2 in 15, which means that for every 15 cm horizontally its rises 2 cm. Calculate the length of the base of the ramp.

Solution

Problem 2 :

A boy who is 1.6 m tall casts a 2.4 m shadow when he stands 8.1 m from the base of an electric light pole. How high above the ground is the light globe ?

Solution

Problem 3 :

A piece of timber leaning against a wall, just touches the top of a fence, as shown. Find how far up the wall the timber reaches.

Solution

Problem 4 :

At the same time as the shadow cast by a vertical 30 cm long ruler is 45 cm long, Rafael’s shadow is 264 cm long.

a) Draw a fully labelled sketch of the situation.

b) Find Rafael’s height ?

Solution

Problem 5 :

A 3.5 m ladder leans on a 2.4 m high fence. One end is on the ground and the other end touches a vertical wall 2.9 m from the ground. How far is the bottom of the ladder from the fence ?

Solution

Problem 6 :

Two surveyors estimate the height of a nearby mill. One stands 5 m away from the other on horizontal ground holding a 3 m stick vertically. The other surveyor finds a “line of sight” to the top of the hill, and observes this line passes the vertical stick at 2.4 m. They measures the distance from the stick to the top of the hill to be 1500 m using laser equipment. How high, correct to the nearest meter, is their estimate of the height of the hill ?

Solution

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