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alukav5142 [94]
3 years ago
8

A triangle has vertices at (1,10), (-5,2), and (7,2). What is its orthocenter. Show your work.

Mathematics
1 answer:
kondor19780726 [428]3 years ago
3 0

Answer:

Question: What is the orthocenter of a triangle with the vertices (-1,2) (5,2) and (2,1)?

The coordinates of point A are (-1,2), point B are (5,2), and point C are (2,1).

The orthocent is the intersection of the three altitudes. An altitude goes from a vertex and is perpendicular to the line containing the opposite side.

In the coordinate plane the equations of the altitudes can be found and then a system of equations can be solved.

Altitude 1. From point C perpendicular to the line containing side AB.

Slope of line AB is 0 (horizontal line), a vertical line is perpendicular to a horizontal line. Thus, the equation of altitude 1 is  x=2 .

Altitude 2. From point B perpendicular to the line containing side AC.

Slope of line AC is  −13 , the slope of a line perpendicular to line AC is 3. The equation of altitude 2 is  y=3x−13  

Altitude 3. From point A perpendicular to the line containing side BC.

Slope of line BC is  13 , the slope of a line perpendicular to line BC is  −3 . The equation of altitude 3 is  y=−3x−1  

The orthocenter is the point where all three altitudes intersect.

x=2  

y=3x−13  

y=−3x−1  

Use substitution to solve the first two equations  y=3(2)−13=−7  

The orthocenter is the point  (2,−7)  

we did not need the third equation, but we can use it as a check, plug the coordinates into the third equation:

−7=−3(2)−1  

−7=−6−1  

−7=−7  it works.

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3 years ago
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Answer:

21440

Step-by-step explanation:

<h2>Simplify:</h2>
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Start by multiplying 7x³ by x² and -5.

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Multiply 8x² by x² and -5.

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Multiply -3 by x² and -5.

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Combine like terms together.

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Rearrange the terms in descending power order.

  • 7x⁵ + 8x⁴ - 35x³ - 43x² + 15  
<h2>Verify (I): </h2>

Substitute x = 5 into the above polynomial.

  • 7(5)⁵ + 8(5)⁴ - 35(5)³ - 43(5)² + 15

Evaluate the exponents first.

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Multiply the terms together.

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Combine the terms together.

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This is the answer when substituting x = 5 into the simplified expression.

<h2>Verify (II):</h2>

Substitute x = 5 into the expression.

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Evaluate the exponents first.

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Multiply the terms in the first bracket next.

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Multiply these two terms together.

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This is the answer when substituting x = 5 into the original (unsimplified) expression.

3 0
3 years ago
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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

The slope is 1 (since (6-4)/(3-1)=2/2=1)

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