A triangle has corners A, B, and C located at #(7 ,3 )#, #(4 ,8 )#, and #(3 , 7 )#, respectively. What are the endpoints and length of the altitude going through corner C?
The end points are
The triangle's corners are
The altitude's duration is
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End points are
Length of altitude is
By solving equations (1) and (2), we can determine D's coordinates.
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To find the altitude going through corner C of the triangle, we first need to determine the equation of the line containing side AB, which will be perpendicular to the altitude passing through corner C. Then, we can find the intersection point of this line with the altitude to determine its endpoints. Finally, we calculate the distance between the intersection point and corner C to find the length of the altitude.

Calculate the slope of the line passing through corners A and B using the formula ( m = \frac{{y_2  y_1}}{{x_2  x_1}} ). [ m_{AB} = \frac{{8  3}}{{4  7}} = \frac{5}{3} = \frac{5}{3} ]

Since the altitude through corner C is perpendicular to side AB, the slope of the altitude is the negative reciprocal of the slope of side AB. Therefore, the slope of the altitude (( m_{\text{altitude}} )) is the negative reciprocal of ( m_{AB} ), which is ( \frac{3}{5} ).

Using the pointslope form of the equation of a line ( y  y_1 = m(x  x_1) ), where ( (x_1, y_1) ) is a point on the line, we can write the equation of the altitude passing through corner C using point C (( (3, 7) )) and the slope ( \frac{3}{5} ): [ y  7 = \frac{3}{5}(x  3) ]

Solve this equation for ( y ) to find the equation of the altitude: [ y = \frac{3}{5}x + 5 ]

Next, we find the intersection point of this line with the line passing through corner C, which is the altitude. We set the equations of the altitude and the line passing through corner C equal to each other and solve for ( x ) and ( y ). [ \frac{3}{5}x + 5 = 7 ] [ \frac{3}{5}x = 2 ] [ x = \frac{10}{3} ]

Substitute ( x = \frac{10}{3} ) into either equation to find ( y ): [ y = \frac{3}{5} \left(\frac{10}{3}\right) + 5 = \frac{18}{5} + 5 = \frac{18}{5} + \frac{25}{5} = \frac{43}{5} ]

So, the intersection point of the altitude with corner C is ( \left(\frac{10}{3}, \frac{43}{5}\right) ).

Finally, calculate the length of the altitude, which is the distance between this intersection point and corner C using the distance formula: [ d = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} ] [ d = \sqrt{\left(\frac{10}{3}  3\right)^2 + \left(\frac{43}{5}  7\right)^2} ] [ d = \sqrt{\left(\frac{1}{3}\right)^2 + \left(\frac{3}{5}\right)^2} ] [ d = \sqrt{\frac{1}{9} + \frac{9}{25}} ] [ d = \sqrt{\frac{25 + 81}{225}} ] [ d = \sqrt{\frac{106}{225}} ] [ d = \frac{\sqrt{106}}{15} ]
So, the length of the altitude going through corner C is ( \frac{\sqrt{106}}{15} ), and its endpoints are ( \left(\frac{10}{3}, \frac{43}{5}\right) ) and ( (3, 7) ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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