Carcass wrote:

In the coordinate system below, ﬁnd the following.

(a) Coordinates of point \(Q\)

(b) Lengths of \(PQ, QR,\) and \(PR\)

(c) Perimeter of \(\triangle PQR\)

(d) Area of \(\triangle PQR\)

(e) Slope, y-intercept, and equation of the line passing through points \(P\) and \(R\)

Attachment:

#GREexcercises In the coordinate system below, ﬁnd the following. (a) Coordinate.jpg

(a) (−2,0) (b) PQ=6, QR=7, PR=\(\sqrt{85}\) (c) \(13+\sqrt{85}\) (d) 21 (e) Slope: -\(\frac{6}{7}\); y-intercept: \frac{30}{7} ; equation of line: \(y=- \frac{6}{7}x+\frac{30}{7}\), or \(7y+6x=30\)

(a) Coordinates of point \(Q\)

Since point Q is on the x-axis, it's y-coordinate must be 0.

Also, since PQ is PARALLEL to the y-axis, we know that points P and Q are the SAME distance from the y-axis.

This means the two points will have the SAME x-coordinate .

So, the coordinates of point Q are (-2, 0)

----------------------------------

(b) Lengths of \(PQ, QR,\) and \(PR\)

PQ is the line segment from (-2, 0) and (-2, 6).

Length of PQ = 6-0 =

6QR is the line segment from (-2, 0) and (5, 0).

Length of QR = 5-(-2) =

7Finally, PR is the hypotenuse of the right triangle.

Let x = the length of PR

Apply the Pythagorean Theorem to get: 6² + 7² = x²

Evaluate: 36 + 49 = x²

So, 85 = x², which means x =

√85----------------------------------

(c) Perimeter of \(\triangle PQR\)

Perimeter = sum of all 3 sides

= 6 + 7 + √85 =

13 + √85----------------------------------

(d) Area of \(\triangle PQR\)

area = (base)(height)/2

= (7)(6)/2

=

21----------------------------------

(e) Slope, y-intercept, and equation of the line passing through points \(P\) and \(R\)

Slope = rise/run =

-6/7Let's write the equation of the line in slope y-intercept form: y = mx + b, where m = slope and b = y-intercept

We get: \(y = -\frac{6}{7}x+b\) (we don't know the value of b yet, but we soon will)

Since the point (5, 0) lies ON the line, we know that x = 5 and y = 0 is a solution to the equation of the line.

So, plug in x = 5 and y = 0 to get: \(0 = -\frac{6}{7}(5)+b\)

Simplify: \(0 = -\frac{30}{7}+b\)

Solve: \(b = \frac{30}{7}\)

This means the y-intercept is

30/7Finally, the equation of the line (in slope y-intercept form) is \(y = -\frac{6}{7}x+\frac{30}{7}\)

Cheers,

Brent

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Brent Hanneson – Creator of greenlighttestprep.com

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