We read your photo as:

Please answer the question and the answers should be in English

English

ScanSolve AI Answer

Answer

(133,83)\left(\frac{13}{3}, \frac{8}{3}\right)

Step-by-step solution

6 steps

To find the coordinates of a point equidistant from three given points, we can set up equations using the distance formula. Let the unknown point be P(x,y)P(x, y) and the given points be A(2,3)A(2, 3), B(4,5)B(4, 5), and C(6,1)C(6, 1). The condition is PA=PB=PCPA = PB = PC, which implies PA2=PB2=PC2PA^2 = PB^2 = PC^2.

  1. Set up the equation PA2=PB2PA^2 = PB^2.

    Using the distance formula squared, (x2−x1)2+(y2−y1)2(x_2 - x_1)^2 + (y_2 - y_1)^2:

    (x−2)2+(y−3)2=(x−4)2+(y−5)2(x - 2)^2 + (y - 3)^2 = (x - 4)^2 + (y - 5)^2

  2. Expand and simplify the equation from Step 1.

    x2−4x+4+y2−6y+9=x2−8x+16+y2−10y+25x^2 - 4x + 4 + y^2 - 6y + 9 = x^2 - 8x + 16 + y^2 - 10y + 25

    Subtract x2x^2 and y2y^2 from both sides:

    −4x−6y+13=−8x−10y+41-4x - 6y + 13 = -8x - 10y + 41

    Rearrange the terms to form a linear equation:

    −4x+8x−6y+10y=41−13-4x + 8x - 6y + 10y = 41 - 13 4x+4y=284x + 4y = 28

    Divide by 4:

    x+y=7(Equation 1)x + y = 7 \quad (\text{Equation 1})

  3. Set up the equation PB2=PC2PB^2 = PC^2.

    (x−4)2+(y−5)2=(x−6)2+(y−1)2(x - 4)^2 + (y - 5)^2 = (x - 6)^2 + (y - 1)^2

  4. Expand and simplify the equation from Step 3.

    x2−8x+16+y2−10y+25=x2−12x+36+y2−2y+1x^2 - 8x + 16 + y^2 - 10y + 25 = x^2 - 12x + 36 + y^2 - 2y + 1

    Subtract x2x^2 and y2y^2 from both sides:

    −8x−10y+41=−12x−2y+37-8x - 10y + 41 = -12x - 2y + 37

    Rearrange the terms:

    −8x+12x−10y+2y=37−41-8x + 12x - 10y + 2y = 37 - 41 4x−8y=−44x - 8y = -4

    Divide by 4:

    x−2y=−1(Equation 2)x - 2y = -1 \quad (\text{Equation 2})

  5. Solve the system of linear equations (Equation 1 and Equation 2).

    We have:

    1. x+y=7x + y = 7
    2. x−2y=−1x - 2y = -1

    Subtract Equation 2 from Equation 1:

    (x+y)−(x−2y)=7−(−1)(x + y) - (x - 2y) = 7 - (-1) x+y−x+2y=8x + y - x + 2y = 8 3y=83y = 8 y=83y = \frac{8}{3}

    Substitute the value of yy into Equation 1:

    x+83=7x + \frac{8}{3} = 7 x=7−83x = 7 - \frac{8}{3} x=213−83x = \frac{21}{3} - \frac{8}{3} x=133x = \frac{13}{3}

  6. State the coordinates of the point.

    The coordinates of the point equidistant from the three given points are (133,83)\left(\frac{13}{3}, \frac{8}{3}\right).

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More English Questions

QAnswer the question . The answer should be in EnglishQPlease solve the questions and the answers should be in EnglishQPlease solve this question and the anwers should be in EnglishQPlease answer in EnglishQAnswer the question in English
See all English questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp