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Geometry

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1. In△ABC, we know that AB=5, BC=9, and CA=7.D is a point on BC such that AD

bisects ∠BAC. What is DC?

2. O is the circumcenter of △ABC. If OB = OC = BC, what is ∠BAC?

3. H is the orthocenter of △ABC. We know AB = AC = BC/sqrt3 What is ∠BHC?

4. E is a point on the side CD of square ABCD. The extensions of AD and BE intersect at F. [DEF] = 8 and [ECB] = 18. Find [ABCD].

5. CD and EF are parallel to AB. They divide the area of △PAB into three equal parts. If PE = 1, what is AC?

Dec 19, 2022
edited by Chickenfingrrss  Dec 19, 2022

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Here are the photos * there isn't a photo for #3; the number correspond to the question numbers.

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Dec 19, 2022
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1. CD = 16/3.

2. Angle BAC = 40 degrees.

Guest Dec 19, 2022
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Let me know how you got that!

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3. Angle BHC = 135 degrees.

4. [ABCD] = 44.

5. AC = 2 - sqrt(2)

Dec 19, 2022
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1)  Solving SSS triangle ABC [5, 7, 9] using The Law of Cosines, we get:

Angle ∠ C =33.557° = 33°33'26″ = 0.586 rad
Angle ∠ B =50.704° = 50°42'13″ = 0.885 rad
Angle ∠ A =95.739° = 95°44'21″ = 1.671 rad

Angle DAC = 95.739 / 2 = 47.8695 degrees

In triangle ACD, angle C = 33.557 degrees

Solving ASA triangle ACD, we get:

Sides: DC = 5.25   AD = 3.913   AC = 7

Dec 19, 2022