Find the area of a circle of radius 4 cm.
1 answer:
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Answer: x = 10
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Explanation:
Check out the attached image. I've basically redrawn your image and added point D. This point is the intersection of the angle bisector and segment BC. The red angle (DAC) and the green angle (DAB) are congruent.
The reason why I added point D is to help with the angle bisector theorem. This theorem states that if we have a triangle ABC, and angle A is bisected (as the diagram shows), then we can set up the following proportion
AB/BD = AC/CD
Note how AB and BD are on the left side of point D, and they form the ratio AB/BD. Similarly, AC and CD are to the right of point D so they form the ratio AC/CD. These two ratios are equal by the angle bisector theorem.
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Let's plug in the given info and then isolate x
AB/BD = AC/CD
8/(x+4) = 12/(2x+1)
8(2x+1) = 12(x+4) ... cross multiply
8(2x)+8(1) = 12(x)+12(4) ... distribute
16x+8 = 12x+48
16x+8-8 = 12x+48-8 ... subtract 8 from both sides
16x = 12x+40
16x-12x = 12x+40-12x ... subtract 12x from both sides
4x = 40
4x/4 = 40/4 ... divide both sides by 4
x = 10
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Check:
8/(x+4) = 12/(2x+1)
8/(10+4) = 12/(2*10+1) ... replace every x with 10
8/(10+4) = 12/(20+1)
8/14 = 12/21
4/7 = 4/7
Answer is confirmed
========================================================
Explanation:
Check out the attached image. I've basically redrawn your image and added point D. This point is the intersection of the angle bisector and segment BC. The red angle (DAC) and the green angle (DAB) are congruent.
The reason why I added point D is to help with the angle bisector theorem. This theorem states that if we have a triangle ABC, and angle A is bisected (as the diagram shows), then we can set up the following proportion
AB/BD = AC/CD
Note how AB and BD are on the left side of point D, and they form the ratio AB/BD. Similarly, AC and CD are to the right of point D so they form the ratio AC/CD. These two ratios are equal by the angle bisector theorem.
---------------------
Let's plug in the given info and then isolate x
AB/BD = AC/CD
8/(x+4) = 12/(2x+1)
8(2x+1) = 12(x+4) ... cross multiply
8(2x)+8(1) = 12(x)+12(4) ... distribute
16x+8 = 12x+48
16x+8-8 = 12x+48-8 ... subtract 8 from both sides
16x = 12x+40
16x-12x = 12x+40-12x ... subtract 12x from both sides
4x = 40
4x/4 = 40/4 ... divide both sides by 4
x = 10
---------------------
Check:
8/(x+4) = 12/(2x+1)
8/(10+4) = 12/(2*10+1) ... replace every x with 10
8/(10+4) = 12/(20+1)
8/14 = 12/21
4/7 = 4/7
Answer is confirmed