1.
[latex]PQRS[/latex] is an isosceles trapezoid.
Chapter 1: Triangles and Circles
Exercises: 1.2 Similar Triangles
Skills
- Identify congruent triangles and find unknown parts #1-6
- Identify similar triangles #7-10
- Find unknown parts of similar triangles #11-20
- Solve problems using proportions and similar triangles #21-26
- Use proportions to relate sides of similar triangles #27-38
Suggested Problems
Problems: #2, 8, 14, 22, 28, 32.
Exercises for 1.2 Similar Triangles
Exercise Group
In Problems 1–4, name two congruent triangles and find the unknown quantities.
2.
3.
[latex]\triangle PRU[/latex] is isosceles.
4.
[latex]\triangle PRU[/latex] is isosceles and [latex]OR=NG[/latex]. Find [latex]\angle RNG[/latex] and [latex]\angle RNO[/latex]
5.
Delbert and Francine want to measure the distance across a stream. They mark point [latex]A[/latex] directly across the stream from a tree at point [latex]T[/latex] on the opposite bank. Delbert walks from point [latex]A[/latex] down the bank a short distance to point [latex]B[/latex] and sights the tree. He measures the angle between his line of sight and the stream bank.
- Draw a figure showing the stream, the tree, and right triangle [latex]ABT[/latex].
- Meanwhile, Francine, who was still standing at point [latex]A[/latex], walks away from the stream at right angles to Delbert’s path. Delbert watches her progress and tells her to stop at point [latex]C[/latex], when the angle between the stream bank and his line of sight to Francine is the same as the angle from the stream bank to the tree. Add triangle [latex]ABC[/latex] to your figure.
- Delbert now measures the distance from point [latex]A[/latex] to Francine at point [latex]C[/latex]. Explain why this distance is the same as the distance across the stream.
6.
If you have a baseball cap, here is another way to measure the distance across a river. Stand at point [latex]A[/latex] directly across the river from a convenient landmark, say a large rock, on the other side. Tilt your head down so that the brim of the cap points directly at the base of the rock, [latex]R[/latex].
- Draw a figure showing the river, the rock, and right triangle [latex]ABR[/latex], where [latex]B[/latex] is the location of your baseball cap on your head.
- Now, without changing the angle of your head, rotate [latex]90^{o}[/latex] and sight along the bank on your side of the river. Have a friend mark the spot [latex]C[/latex] on the ground where the brim of your cap points. Add triangle [latex]ABC[/latex] to your figure.
- Finally, you can measure the distance from point [latex]A[/latex] to point [latex]C[/latex]. Explain why this distance is the same as the distance across the river.
Exercise Group
For Problems 7–10, decide whether the triangles are similar, and explain why or why not.
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Exercise Group
Assume the triangles in Problems 11–14 are similar. Solve for the variables. (Figures are not drawn to scale.)
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Exercise Group
In Problems 15–20, use properties of similar triangles to solve for the variable.
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Exercise Group
For Problems 21–26, use properties of similar triangles to solve.
21.
A rock-climber estimates the height of a cliff she plans to scale as follows. She places a mirror on the ground so that she can just see the top of the cliff in the mirror while she stands straight.
The angles [latex]and [latex]\angle 2[/latex] formed by the light rays are equal, as shown in the figure. She then measures the distance to the mirror ([latex]2[/latex] feet) and the distance from the mirror to the base of the cliff ([latex]56[/latex] feet). If she is [latex]5[/latex] feet [latex]6[/latex] inches tall, how high is the cliff?
22.
Edo estimates the height of the Washington Monument as follows. He notices that he can see the reflection of the top of the monument in the reflecting pool. He is feet from the tip of the reflection, and that point is [latex]1080[/latex] yards from the base of the monument, as shown below. From his physics class, Edo knows that the angles marked are equal. If Edo is [latex]6[/latex] feet tall, what is his estimate for the height of the Washington Monument?
23.
In the sixth century BC, the Greek philosopher and mathematician Thales used similar triangles to measure the distance to a ship at sea. Two observers on the shore at points [latex]A[/latex] and [latex]B[/latex] would sight the ship and measure the angles formed, as shown in figure (a). They would then construct a similar triangle as shown in figure (b), with the same angles at [latex]A[/latex] and [latex]B[/latex] and measure its sides. (This method is called triangulation.) Use the lengths given in the figures to find the distance from observer to the ship.
24.
The Capilano Suspension Bridge is a footbridge that spans a [latex]230[/latex]-foot gorge north of Vancouver, British Columbia. Before crossing the bridge, you decide to estimate its length.
You walk [latex]100[/latex] feet downstream from the bridge and sight its far end, noting the angle formed by your line of sight, as shown in figure (a). You then construct a similar right triangle with a 2-centimeter base, as shown in figure (b). You find that the height of your triangle is [latex]8.98[/latex] centimeters. How long is the Capilano Suspension Bridge?
25.
A conical tank is [latex]12[/latex] feet deep, and the diameter of the top is [latex]8[/latex] feet. If the tank is filled with water to a depth of [latex]7[/latex] feet, as shown in the figure at right, what is the area of the exposed surface of the water?
26.
To measure the distance [latex]EC[/latex] across the lake shown in the figure at right, stand at [latex]A[/latex] and sight point [latex]C[/latex] across the lake, then mark point [latex]B[/latex]. Then sight to point [latex]E[/latex] and mark point [latex]D[/latex] so that [latex]DB[/latex] is parallel to [latex]EC[/latex]. If [latex]AD = 25[/latex] yards, [latex]AE = 60[/latex] yards, and [latex]BD = 30[/latex] yards, how wide is the lake?
Exercise Group
In Problems 27–28, the pairs of triangles are similar. Solve for [latex]y[/latex] in terms of [latex]x[/latex]. (The figures are not drawn to scale.)
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Exercise Group
For Problems 29–34, use properties of similar triangles to solve for the variable.
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Exercise Group
In Problems 35–38, solve for [latex]y[/latex] in terms of [latex]x[/latex].
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39.
Triangle [latex]ABC[/latex] is a right triangle, and [latex]AD[/latex] meets the hypotenuse [latex]BC[/latex] at a right angle.
- If [latex]\angle ACB = 20^{o}[/latex] find [latex]\angle B, \angle CAD[/latex], and [latex]\angle DAB[/latex]
- Find two triangles similar to [latex]\triangle ABC[/latex]. List the corresponding sides in each of the triangles.
40.
Here is a way to find the distance across a gorge using a carpenter’s square and a five-foot pole. Plant the pole vertically on one side of the gorge at point [latex]A[/latex] and place the angle of the carpenter’s square on top of the pole at point [latex]B[/latex], as shown in the figure. Sight along one side of the square so that it points to the opposite side of the gorge at point [latex]P[/latex]. Without moving the square, sight along the other side and mark point [latex]Q[/latex]. If the distance from [latex]Q[/latex] to [latex]A[/latex] is six inches, calculate the width of the gorge. Explain your method.
