Answer:
![\sf R=\left(3, -\dfrac{5}{4}\right)](https://tex.z-dn.net/?f=%5Csf%20R%3D%5Cleft%283%2C%20-%5Cdfrac%7B5%7D%7B4%7D%5Cright%29)
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
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<u>Distance between two points</u>
![\sf d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=%5Csf%20d%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
(where (x₁, y₁) and (x₂, y₂) are the two points)
Use the <u>distance formula</u> 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}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20%5Csf%20PR%20%26%20%3D%5Csf%20%5Csqrt%7B%283-1%29%5E2%2B%28b-5%29%5E2%7D%5C%5C%20%26%20%3D%20%5Csf%20%5Csqrt%7B4%2B%28b-5%29%5E2%7D%20%5Cend%7Baligned%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20%5Csf%20QR%20%26%20%3D%5Csf%20%5Csqrt%7B%283-%28-2%29%29%5E2%2B%28b-3%29%5E2%7D%5C%5C%20%26%20%3D%20%5Csf%20%5Csqrt%7B25%2B%28b-3%29%5E2%7D%20%5Cend%7Baligned%7D)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20%5Csf%20PR%20%26%20%3D%20%5Csf%20QR%20%5C%5C%5Csf%20%5Csqrt%7B4%2B%28b-5%29%5E2%7D%20%26%20%3D%20%5Csf%20%5Csqrt%7B25%2B%28b-3%29%5E2%7D%5C%5C%5Csf%204%2B%28b-5%29%5E2%20%26%20%3D%20%5Csf%2025%2B%28b-3%29%5E2%5C%5C%5Csf%204%2Bb%5E2-10b%2B25%20%26%20%3D%20%5Csf%2025%20%2B%20b%5E2-6b%2B9%5C%5C%5Csf%20b%5E2-10b%2B29%20%26%20%3D%20%5Csf%20b%5E2%20-6b%20%2B34%5C%5C%5Csf%20-10b%2B29%20%26%20%3D%20%5Csf%20-6b%20%2B%2034%5C%5C%5Csf%20-4b%20%26%20%3D%20%5Csf%205%5C%5C%5Csf%20b%20%26%20%3D%20-%5Cdfrac%7B5%7D%7B4%7D%5Cend%7Baligned%7D)
Substitute the found values of a and b to find the coordinates of R:
![\sf R=(a,b)=\left(3, -\dfrac{5}{4}\right)](https://tex.z-dn.net/?f=%5Csf%20R%3D%28a%2Cb%29%3D%5Cleft%283%2C%20-%5Cdfrac%7B5%7D%7B4%7D%5Cright%29)
Learn more about the distance formula here:
brainly.com/question/28144723