Benny’s age is currently 200% of his sister Jenny’s age. What percent of Benny’s age will Jenny’s age
1 answer:
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9514 1404 393
Answer:
not tangent
Step-by-step explanation:
The triangle has side lengths 5, 12, 14. You can compute the "form factor" for the triangle:
a^2 +b^2 -c^2 . . . . . . . . where 'c' is the longest side
= 5^2 +12^2 -14^2
= 25 +144 -196 = -27
The negative value tells you this triangle is obtuse. If AB were a tangent, it would be perpendicular to the radius—the triangle would be a right triangle.
If you compare this calculation to the Pythagorean theorem, you see that the length AB is longer than the length √(25+144) = 13 that is necessary for the triangle to be a right triangle. That means the angle at A is greater than 90°.
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<em>Additional comment</em>
This "form factor" calculation is part of the calculation you would do using the Law of Cosines to determine the largest angle. The sign of the "form factor" tells you the sign of the cosine of the angle. Angles whose cosine is negative are greater than 90°. A positive "form factor" indicates an acute triangle.
I find computing the "form factor" in this way makes interpretation of the result fairly easy. For me, it eliminates the confusion I had when the numbers of the Pythagorean theorem didn't add up. For me, it was too much work to figure whether the triangle was acute or obtuse, and I often got it wrong.
For those interested, the angle measure is arccos((a^2+b^2-c^2)/(2ab)). Here, that's arccos(-27/(2·5·12)) ≈ 103.003°.
Answer:
First Case:
Second Case:
Step-by-step explanation:
We know that the first three terms of an arithmetic series are:
Since this is an arithmetic sequence, each subsequent term is <em>d</em> more than the previous term, where <em>d</em> is our common difference.
Therefore, we can write the second term as;
And, likewise, for the third term:
Let's solve for <em>d</em> for each of the equations.
Subtracting in the first equation yields:
And for the second equation:
To avoid fractions, let's multiply the first equation by 2. Hence:
Therefore:
Simplifying yields:
Solve for <em>p</em>. We can factor:
Factor:
Grouping:
Zero Product Property:
Then, we can use the second equation to solve for <em>d</em>. So:
Substituting:
So, for the first case, <em>p</em> is 5/2 and <em>d</em> is -2.
Likewise, for the second case:
So, for the second case, <em>p </em>is -5/4, and <em>d</em> is 7/4.
By using the values, we can determine our series.
For Case 1, we will have:
17, 15, 13.
For Case 2, we will have:
-11/2, -15/4, -2.