Convert the polar equation to rectangular coordinates. (Use variables x and y as needed.)

1 answer:

8 0

The standard conversion toolbox:
x=rcos(θ)
y=rsin(θ)

Here r=7csc(θ)
so
x=7csc(θ)*cos(θ)=7cos(θ)/sin(θ)=7cot(<span>θ)
y=7csc(</span>θ)*sin(θ)=7sin(θ)/sin(<span>θ) = 7
therefore the rectangular coordinates are (7cot(</span><span>θ), 7)</span>

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kolezko [41]

<h3>Proof:</h3>

Refer to the attached figure.

There are a couple of ways you can go at this. One is to show the sums of the marked angles are the same, hence ∠B ≅ ∠D. Instead, we're going to show that ΔABD ≅ ΔCDB, hence ∠A ≅ ∠C.

1. AB║DC and BC║AD . . . . given

2. BD is a transversal to both AB║DC and BC║AD . . . . given

3. ∠CBD ≅ ∠ADB . . . . alternate interior angles where a transversal crosses parallel lines are congruent

4. ∠CDB ≅ ∠ABD . . . . alternate interior angles where a transversal crosses parallel lines are congruent

5. BD ≅ BD . . . . reflexive property of congruence

6. ΔABD ≅ ΔCDB . . . . ASA postulate

7. ∠A ≅ ∠C . . . . CPCTC

∠A and ∠C are opposite angles of parallelogram ABCD, so we have shown what you want to have shown.