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To construct a triangle when two sides and the angle opposite one of them is given. |
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To construct a triangle when two sides and the angle opposite one of them is given. |
Given:  KAL,
side AC, and side BC opposite KAL |
Required: To construct  ABC
with KAL,
side AC, and side BC. |
| Procedure: |
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 Use
the straightedge tool and lay off a base segment AX. |
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Construct YAX
congruent to KAL
(help) |
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Use the circle by center + radius command under the
construct menu to construct a circle with center A and radius congruent
to segment AC. |
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Label the intersection of the circle with ray AY point C. |
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Use the circle by center + radius command under the
construct menu to construct a circle with center C and radius congruent
to segment BC. |
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Label the intersection of the circle with ray AX point B. |
 ABC
is the required . |
| Proof: |
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Consider the various cases discussed below. |
| Sketch: By adjusting the dimensions of the given parts three cases emerge. As you move the points watch the actual measures change. Can you duplicate all the possible cases? |
| Case I Drag point K until KAL
is obtuse. |
| a. If BC < AC, there is no solution. |
| b. If BC = AC, there is no solution. |
| c. If BC > AC, there is one solution. |
| Case II Drag point K until KAL
is a right . |
| a. If BC < AC, there is no solution. |
| b. If BC = AC, there is no solution. |
| c. If BC > AC, there is one solution. |
| Case III Drag point K until KAL
is acute. |
a. If BC <  distance
from C to AX, there is no solution. |
| b. If BC = distance
from C to AX, there is 1 solution. |
| c. If BC > distance
from C to AX but < AC, there are 2 solutions. |
| d. If BC = AC, there is 1 solution. |
| e. If BC > AC, there is 1 solution. |
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This page uses JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright © 1990-2001 by KCP Technologies, Inc. Licensed only for non-commercial use. triangle with two sides and angle opposite one |
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