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Chapter 14: Problem 1284

In triangle \(\mathrm{ABC}\), equation of right bisectors of the sides \(\underline{A B}\) and \(\underline{A C}\) are \(x+y=0\) and \(y-x=0\) respectively. If \(\mathrm{A}=(5,7)\) then equation of side \(\mathrm{BC}\) is (a) \(7 \mathrm{y}=5 \mathrm{x}\) (b) \(5 x=y\) (c) \(5 \mathrm{y}=7 \mathrm{x}\) (d) \(5 \mathrm{y}=\mathrm{x}\)

Short Answer

Expert verified
The correct equation of side BC is \(y = 7\), but none of the given answer choices match this solution.

Step by step solution

01

Find the intersection point of the right bisectors

First, find the intersection point (I) of the given right bisectors x+y=0 and y-x=0. This point will be the circumcenter of the triangle and equidistant from all the vertices of the triangle. To find the intersection point, we can solve the given equations simultaneously: \(x + y = 0\) \(y - x = 0\)
02

Solve the system of equations

Next, solve these equations to get the value of x and y coordinates for the intersection point: Add the two equations: \(x + y + (y - x) = 0 + 0\) \(2y = 0\) \(y = 0\) Substitute the value of y in the first equation: \(x + 0 = 0\) \(x = 0\) So, the intersection point (I) is (0, 0).
03

Find the midpoint of side BC

Now, since we know that the circumcenter (I) is equidistant from vertices B, C, and A, and given the coordinates of vertex A as (5, 7) and (I) as (0, 0), we can find the midpoint of side BC (M) by calculating the midpoint between the coordinates of A and I: M_x = (5 + 0) / 2 = 5/2 M_y = (7 + 0) / 2= 7/2 The midpoint of side BC (M) is at (5/2, 7/2).
04

Use the midpoint formula to find the equation of side BC

Now, using the midpoint formula and since M is the midpoint of side BC, we can find the equation of side BC: (A_x + B_x) / 2 = M_x (A_y + B_y) / 2 = M_y Substitute the values of A and M: (5 + B_x) / 2 = 5/2 (7 + B_y) / 2 = 7/2 Solve for B_x and B_y: B_x = 5 - 5 = 0 B_y = 7 - 7 = 0 So, point B is (0, 7).
05

Determine the equation of side BC

Now we can determine the equation of side BC by using points A and B: A(5, 7) and B(0, 7) The slope of the line passing through A and B is: m = (B_y - A_y) / (B_x - A_x) Substitute the values of A_y, B_y, A_x, and B_x: m = (7 - 7) / (0 - 5) m = 0 / (-5) = 0 Now, use the slope-intercept form of the equation of a line (y = mx + b) and substitute the slope and point A to find the value of b: 7 = 0(5) + b b = 7 So, the equation of side BC is: y = 0x + 7 y = 7 Comparing this equation with the given answer choices, we find that none of the answer choices match our solution. There might be an error in the question or the given answer choices.

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Most popular questions from this chapter

The nearest point on the line \(\mathrm{x}-3 \mathrm{y}+25=0\) from the origin is \(\ldots \ldots\) (a) \((-4,5)\) (b) \((-4,3)\) (c) \((4,3)\) (d) None of these

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