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Answer:
see attached
Step-by-step explanation:
Dilation is the transformation that changes the size of something. When you choose ×110% on a copy machine, you are making all the dimensions in the copy equal to 1.1 times the dimensions of the original, for example. That is dilation by a factor of 1.1, or 110%.
Here, you're asked to dilate triangle ABC by a factor of 3. That means all of the dimensions on the dilated figure will be 3 times their length on the original.
The center of dilation is point P. That means all the distances from point P are multiplied by 3.
In the attached figure, a gray arrow has been drawn from point P to each of the points A, B, and C. Then two more arrows of the same length are drawn in the same direction. The total distance at the end of the 3 arrows is 3 times the original distance from point P. These are the points A', B', and C' you are asked to draw. Connecting these points gives the dilated triangle A'B'C'.
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You could think of point P (the center of dilation) as having coordinates (0, 0). Then points A, B, and C would have coordinates A(-4, 0), B(0, -3), and C(-4, -3). To dilate these points by a factor of 3, multiply each coordinate value by 3: A'(-12, 0), B'(0, -9), C'(-12, -9). You will see that these are the coordinates of the vertices of the dilated triangle.
Answer:
Step-by-step explanation:
please note that to find but α+β+γ in other words the sum of α,β and γ not α,β and γ individually so it's not an equation
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we want to find all possible values of α+β+γ when <u>tanα+tanβ+tanγ = tanαtanβtanγ</u><u> </u>to do so we can use algebra and trigonometric skills first
cancel tanγ from both sides which yields:
factor out tanγ:
divide both sides by tanαtanβ-1 and that yields:
multiply both numerator and denominator by-1 which yields:
recall angle sum indentity of tan:
let α+β be t and transform:
remember that tan(t)=tan(t±kπ) so
therefore <u>when</u><u> </u><u>k </u><u>is </u><u>1</u> we obtain:
remember Opposite Angle identity of tan function i.e -tan(x)=tan(-x) thus
recall that if we have common trigonometric function in both sides then the angle must equal which yields:
isolate -α-β to left hand side and change its sign:
<u>when</u><u> </u><u>i</u><u>s</u><u> </u><u>0</u>:
likewise by Opposite Angle Identity we obtain:
recall that if we have common trigonometric function in both sides then the angle must equal therefore:
isolate -α-β to left hand side and change its sign:
and we're done!