Question

Please help, the question below

Answer 1

Answer:

$\sf R=\left(3, -\dfrac{5}{4}\right)$

Step-by-step explanation:

Given:

• P = (1, 5)
• Q = (-2, 3)
• R = (a, b)
• R lies on line x = 3
• PR = QR

If point R lies on the line x = 3, then the x-value of point R is 3.

⇒ a = 3

Distance between two points

$\sf d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

(where (x₁, y₁) and (x₂, y₂) are the two points)

Use the distance formula to derive equations for PR and QR.

Let (x₁, y₁) = P = (1, 5)

Let (x₂, y₂) = R = (3, b)

$\begin{aligned} \sf PR & =\sf \sqrt{(3-1)^2+(b-5)^2}\\ & = \sf \sqrt{4+(b-5)^2} \end{aligned}$

Let (x₁, y₁) = Q = (-2, 3)

Let (x₂, y₂) = R = (3, b)

$\begin{aligned} \sf QR & =\sf \sqrt{(3-(-2))^2+(b-3)^2}\\ & = \sf \sqrt{25+(b-3)^2} \end{aligned}$

As PR = QR, equate the derived equations and solve for b:

$\begin{aligned} \sf PR & = \sf QR \\\sf \sqrt{4+(b-5)^2} & = \sf \sqrt{25+(b-3)^2}\\\sf 4+(b-5)^2 & = \sf 25+(b-3)^2\\\sf 4+b^2-10b+25 & = \sf 25 + b^2-6b+9\\\sf b^2-10b+29 & = \sf b^2 -6b +34\\\sf -10b+29 & = \sf -6b + 34\\\sf -4b & = \sf 5\\\sf b & = -\dfrac{5}{4}\end{aligned}$

Substitute the found values of a and b to find the coordinates of R:

$\sf R=(a,b)=\left(3, -\dfrac{5}{4}\right)$

Learn more about the distance formula here:

brainly.com/question/28144723

Answer 2

Answer:

R = (3, - 1.25 )

Step-by-step explanation:

using the distance formula

d = $\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }$

given R lies on the line x = 3 then x- coordinate of R is 3 , so

R = (a, b ) = (3, b )

given PR = QR , then calculate PR and QR and equate them

P (1, 5 ) R (3, b )

PR = $\sqrt{(3-1)^2+(b-5)^2}$ = $\sqrt{2^2+(b-5)^2}$ = $\sqrt{4+(b-5)^2}$

Q (- 2, 3 ) R (3, b )

QR = $\sqrt{(3-(-2))^2+((b-3)^2}$ = $\sqrt{(3+2)^2+(b-3)^2}$ = $\sqrt{5^2+(b-3)^2}$, that is

QR = $\sqrt{25+(b-3)^2}$

then equating PR = QR

$\sqrt{4+(b-5)^2}$ = $\sqrt{25+(b-3)^2}$ ( square both sides to clear the radicals )

4 + (b - 5 )² = 25 + (b - 3 )² ← expand factors using FOIL

4 + b² - 10b + 25 = 25 + b² - 6b + 9

b² - 10b + 29 = b² - 6b + 34 ( subtract b² from both sides )

- 10b + 29 = - 6b + 34 ( add 6b to both sides )

- 4b + 29 = 34 ( subtract 29 from both sides )

- 4b = 5 ( divide both sides by - 4 )

b = - 1.25

then

R = (a, b ) = (3, - 1.25 )