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"Solution to Right Triangle" PPT Courseware 2
There are four basic types of solutions to right triangles:
(1) The hypotenuse and the right angle are known;
(2) Two right angle sides are known;
(3) The hypotenuse and an acute angle are known;
(4) Given a right angled side and an acute angle, the solution steps are as follows:
1. (3 points) In △ABC, ∠C=90°, b=3, c=23, then ∠A=________, ∠B=________.
2. (3 points) (2013•Jingzhou) In △ABC, ∠A=120°, AB=4, AC=2, then the value of sin B is ()
A.5714
B.35
C.217
D.2114
3． (3 points) (2013·Lanzhou) In △ABC, a, b, and c are the opposite sides of ∠A, ∠B, and ∠C respectively. If a2＋b2=c2, then the following conclusion is correct ()
A． csin A＝a
B． bcos B＝c
C. atan A＝b
D. ctan B＝b
4. (6 points) As shown in the figure, in Rt△ABC, ∠C=90°, BC=2, AC=4, find the lengths of ∠A, ∠B and AB.
【Integrated use】
18. (16 points) It is known that in △ABC, ∠A is an acute angle, AB=c, BC=a, CA=b.
(1) When ∠A=30°, b=6, c=3, S△ABC=______, 12bc·sin A=________;
(2) When ∠A=45°, b=6, c=3, S△ABC=______, 12bc·sin A=________;
(3) When ∠A=60°, b=4, c=3, S△ABC=______, 12bc·sin A=________;
(4) Based on the answers to questions (1), (2), and (3), guess the relationship between S△ABC and 12bc·sin A, and give the proof.
18. (1) As shown in Figure ①, CD⊥AB is drawn through point C, and the vertical foot is point D. In Rt△ADC, CD=AC·sin A=b·sin 30°=6×12=3, so S △ABC＝12AB·CD＝12×3×3＝92, and 12bc·sin 30°＝92
(2) As shown in Figure ②, similar to (1), S△ABC=12×6×3×22=922, and 12bc·sin 45°=922
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