2.7 Solving Problems Involving More than One Right Triangle
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1 2.7 Solving roblems Involving ore than One Right Triangle OCUS Use trigonometric ratios to solve problems that involve more than one right triangle. When a problem involves more than one right triangle, we can use information from one triangle to solve the other triangle. xample 1 Solving a roblem with Two Triangles ind the length of C to the nearest tenth of a centimetre. A 22.9 cm C To solve a right triangle we must know: the lengths of two sides, or the length of one side and the measure of one acute angle Solution irst use A to find the length of. sin A hypotenuse sin A A sin sin A 22.9 cm 26 o not clear the calculator screen. Side is common to both triangles. In C, find the length of C. sin C sin C sin 49 hypotenuse C C C sin C sin 49 C C is about 13.3 cm long cm 49 C Copyright 2011 earson Canada Inc. 117
2 Check 1. ind the measure of to the nearest degree cm 9.6 cm Use to find the length of. Use the sine ratio cm Side is common to both triangles. sin sin sin In, use the ratio to find. 9.6 cm The measure of is about. The angle of elevation is the angle between the horizontal and a person s line of sight to an object above. object angle of elevation horizontal Copyright 2011 earson Canada Inc.
3 xample 2 Solving a roblem Involving Angle of levation Jason is lying on the ground midway between two trees, 100 m apart. The angles of elevation of the tops of the trees are 13 and 18. ow much taller is one tree than the other? ive the answer to the nearest tenth of a metre. N K 13 Tree 1 18 Tree 2 J 100 m Solution Jason is midway between the trees. So, the distance from Jason to the base of each tree is: Use JK to find the length of JK. 100 m 2 50 m Tree 1 K J tan JK tan Substitute: 13 and J 50 J tan tan 13 JK JK Use N to find the length of N m 50 m tan N tan Substitute: 18 and 50 tan JK 50 N tan 18 N adjacent N Tree 2 adjacent N To find how much taller one tree is than the other, subtract: m m m One tree is about 4.7 m taller than the other. We know 13. JK is. J is adjacent to. Use the tangent ratio. We know 18. N is. is adjacent to. Use the tangent ratio Copyright 2011 earson Canada Inc. 119
4 Check 1. The angle of elevation of the top of a tree, T, is 27. rom the same point on the ground, the angle of elevation of a hawk,, flying directly above the tree is 43. The tree is 12.7 m tall. ow high is the hawk above the ground? ive your answer to the nearest tenth of a metre T 12.7 m We want to find the length of. T Use T to find the length of. Use the tangent ratio m tan tan Substitute: and tan In, use the tangent ratio to find. 43 The hawk is about above the ground. The angle of depression is the angle between the horizontal and a person s line of sight to an object below. horizontal angle of depression object Copyright 2011 earson Canada Inc.
5 xample 3 Solving a roblem Involving Angle of epression rom a small plane, V, the angle of depression of a sailboat is 21. The angle of depression of a ferry on the other side of the plane is 52. The plane is flying at an altitude of 1650 m. ow far apart are the boats, to the nearest metre? V m U sailboat Solution X W ferry We want to find the length of UW. The angle of depression of the sailboat is 21. So, in UVX, V 90 21, or 69. Use UVX to find the length of UX. tan V UX tan V Substitute: V 69 and VX 1650 VX UX tan tan 69 UX adjacent UX U V m X We know V 69. UX is V. VX is adjacent to V. So, use the tangent ratio. The angle of depression of the ferry is 52. So, V in VWX is: 90 52, or 38. Use VWX to find the length of WX. tan V WX tan V Substitute: V 38 and VX 1650 VX tan 38 WX tan 38 WX adjacent WX To find the distance between the boats, add: m m m The boats are about 5588 m apart. V m X W ferry We know V 38. WX is V. VX is adjacent to V. So, use the tangent ratio Copyright 2011 earson Canada Inc. 121
6 Check 1. This diagram shows a falcon,, on a tree, with a squirrel, S, and a chipmunk, C, on the ground. rom the falcon, the angles of depression of the animals are 36 and 47. ow far apart are the animals on the ground to the nearest tenth of a metre? m C S We want to find the length of CS. CS S C The angle of depression of the squirrel is. So, in S is: 90, or. 15 m 36 Use S to find the length of S. tan S tan tan S The angle of depression of the chipmunk is. So, in C is: 90, or. Use C to find the length of C m C C To find the distance between the animals, subtract: The animals on the ground are about apart Copyright 2011 earson Canada Inc.
7 ractice 1. ind the measure of C to the nearest degree. C Use A to find the length of. A cm 22.5 cm A cm Use the tangent ratio. tan A tan A tan In C, use the ratio to find C. C 22.5 cm The measure of C is about Copyright 2011 earson Canada Inc. 123
8 2. Two guy wires support a flagpole,. The first wire is 11.2 m long and has an angle of inclination of 39. The second wire has an angle of inclination of 47. ow tall is the flagpole to the nearest tenth of a metre? 11.2 m Recall that the angle the wire makes with the ground is called the angle of inclination We want to find the length of. Use to find the length of. Use the cosine ratio m Side is common to both triangles. cos 39 cos cos In, use the ratio to find the length of. 47 The flagpole is about tall Copyright 2011 earson Canada Inc.
9 3. A mountain climber is on top of a mountain that is 680 m high. The angles of depression of two points on sides of the mountain are 48 and 32. ow long would a tunnel be that runs between the two points? ive your answer to the nearest metre m N We want to find the length of N. The angle of depression of point is. So, in is: 90, or m Use to find the length of. Use the ratio. The angle of depression of point N is. So, in N is: 90, or. Use N to find the length of N. Use the ratio. 680 m 32 N N The length of the tunnel is: N The tunnel would be about long Copyright 2011 earson Canada Inc. 125
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