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kkurt [141]
2 years ago
6

Lizzy is running a lemonade stand. She has a special price of $3 for 2 glasses of lemonade. She starts the day with no money. De

termine the missing values in the table. The number of glasses of lemonade Lizzie has sold is represented by x, and the amount of money she has earned, in dollars, is represented by y.
Mathematics
1 answer:
sveticcg [70]2 years ago
5 0

Answer:

ercgtbvyuh7

Step-by-step explanation:

vhcrc6vthby7uj7vt6crgtvy

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In right △ABC, the altitude CH to the hypotenuse AB intersects angle bisector AL in point D. Find the sides of △ABC if AD = 8 cm
tangare [24]

Answer:

AC=8\sqrt{3}\ cm\\ \\AB=16\sqrt{3}\ cm\\ \\BC=24\ cm

Step-by-step explanation:

Consider right triangle ADH ( it is right triangle, because CH is the altitude). In this triangle, the hypotenuse AD = 8 cm and the leg DH = 4 cm. If the leg is half of the hypotenuse, then the opposite to this leg angle is equal to 30°.

By the Pythagorean theorem,

AD^2=AH^2+DH^2\\ \\8^2=AH^2+4^2\\ \\AH^2=64-16=48\\ \\AH=\sqrt{48}=4\sqrt{3}\ cm

AL is angle A bisector, then angle A is 60°. Use the angle's bisector property:

\dfrac{CA}{CD}=\dfrac{AH}{HD}\\ \\\dfrac{CA}{CD}=\dfrac{4\sqrt{3}}{4}=\sqrt{3}\Rightarrow CA=\sqrt{3}CD

Consider right triangle CAH.By the Pythagorean theorem,

CA^2=CH^2+AH^2\\ \\(\sqrt{3}CD)^2=(CD+4)^2+(4\sqrt{3})^2\\ \\3CD^2=CD^2+8CD+16+48\\ \\2CD^2-8CD-64=0\\ \\CD^2-4CD-32=0\\ \\D=(-4)^2-4\cdot 1\cdot (-32)=16+128=144\\ \\CD_{1,2}=\dfrac{-(-4)\pm\sqrt{144}}{2\cdot 1}=\dfrac{4\pm 12}{2}=-4,\ 8

The length cannot be negative, so CD=8 cm and

CA=\sqrt{3}CD=8\sqrt{3}\ cm

In right triangle ABC, angle B = 90° - 60° = 30°, leg AC is opposite to 30°, and the hypotenuse AB is twice the leg AC. Hence,

AB=2CA=16\sqrt{3}\ cm

By the Pythagorean theorem,

BC^2=AB^2-AC^2\\ \\BC^2=(16\sqrt{3})^2-(8\sqrt{3})^2=256\cdot 3-64\cdot 3=576\\ \\BC=24\ cm

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