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Q. 43

Expert-verified

Geometry

Found in: Page 527

Book edition Student Edition

Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages 227 pages

ISBN 9780395977279

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Short Answer

It is known that triangle GHM is isosceles. G is point (-2, -3), H is point (-2, 7), and the x-coordinate of M is 4. Find all five possible values for the y-coordinate of M.

The five possible values for the y-coordinate of M are $- 11, 5, 15, - 1, 2$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step-1 – Given

The given points are $G (- 2, - 3), H (- 2, 7)$

Step-2 – To determine

We have to find all possible values for the y-coordinate of M.

Step-3 – Calculation

Let us assume, the y-coordinate of M = y.

So, the coordinate of M is of the form $M (4, y)$ .

We’ll use the distance formula to find the lengths GH, GM and HM.

$d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} [d = distance] So.$

$\begin{array}{l} G H = \sqrt{{(- 2 - (- 2))}^{2} + {(7 - (- 3))}^{2}} [since, G (- 2, - 3) and H (- 2, 7)] \\ G H = \sqrt{{(- 2 + 2)}^{2} + {(7 + 3)}^{2}} \\ G H = \sqrt{{(0)}^{2} + {(10)}^{2}} \\ G H = \sqrt{0 + 100} \\ G H = \sqrt{100} \\ G H = 10 \end{array}$

$\begin{array}{l} G M = \sqrt{{(4 - (- 2))}^{2} + {(y - (- 3))}^{2}} [since, G (- 2, - 3) and M (4, y)] \\ G M = \sqrt{{(4 + 2)}^{2} + {(y + 3)}^{2}} \\ G M = \sqrt{{(6)}^{2} + {(y + 3)}^{2}} \\ G M = \sqrt{36 + y^{2} + 2 (y) (3) + {(3)}^{2}} \\ G M = \sqrt{36 + y^{2} + 6 y + 9} \\ G M = \sqrt{y^{2} + 6 y + 45} \end{array}$

$\begin{array}{l} H M = \sqrt{{(4 - (- 2))}^{2} + {(y - 7)}^{2}} [since, H (- 2, 7) and M (4, y)] \\ H M = \sqrt{{(4 + 2)}^{2} + {(y - 7)}^{2}} \\ H M = \sqrt{{(6)}^{2} + {(y - 7)}^{2}} \\ H M = \sqrt{36 + y^{2} - 2 (y) (7) + {(7)}^{2}} \\ H M = \sqrt{36 + y^{2} - 14 y + 49} \\ H M = \sqrt{y^{2} - 14 y + 85} \end{array}$

For an isosceles triangle:

${(G H)}^{2} = {(G M)}^{2}$

Plug the values of GH and GM in the above equation:

$\begin{array}{l} {(10)}^{2} = {(\sqrt{y^{2} + 6 y + 45})}^{2} \\ 100 = y^{2} + 6 y + 45 \\ y^{2} + 6 y + 45 - 100 = 0 \\ y^{2} + 6 y - 55 = 0 \\ y^{2} + (11 y - 5 y) - 55 = 0 [write, 6 y = (11 y - 5 y)] \\ y^{2} + 11 y - 5 y - 55 = 0 \\ y (y + 11) - 5 (y + 11) = 0 [\begin{array}{l} the first and second term has a common factor, y and \\ the third and fourth term has a common factor,5 \end{array}] \\ (y + 11) (y - 5) = 0 \\ (y + 11) = 0 or (y - 5) = 0 \\ y = - 11 or y = 5 \end{array}$

${(G H)}^{2} = {(H M)}^{2}$

Plug the values of GH and HM in the above equation:

$\begin{array}{l} {(10)}^{2} = {(\sqrt{y^{2} - 14 y + 85})}^{2} \\ 100 = y^{2} - 14 y + 85 \\ y^{2} - 14 y + 85 - 100 = 0 \\ y^{2} - 14 y - 15 = 0 \\ y^{2} - (15 y - y) - 15 = 0 [write, 14 y = (15 y - y)] \\ y^{2} - 15 y + y - 15 = 0 \\ y (y - 15) + 1 (y - 15) = 0 [\begin{array}{l} the first and second term has a common factor, y and \\ the third and fourth term has a common factor,1 \end{array}] \\ (y - 15) (y + 1) = 0 \\ (y - 15) = 0 or (y + 1) = 0 \\ y = 15 or y = - 1 \end{array}$

${(G M)}^{2} = {(H M)}^{2}$

Plug the values of GM and HM in the above equation:

$\begin{array}{l} {(\sqrt{y^{2} + 6 y + 45})}^{2} = {(\sqrt{y^{2} - 14 y + 85})}^{2} \\ y^{2} + 6 y + 45 = y^{2} - 14 y + 85 \\ 6 y + 45 + 14 y - 85 = 0 \\ 20 y - 40 = 0 \\ 20 y = 40 \\ y = 2 [divide both side by 20] \end{array}$

So, the five possible values for the y-coordinate of M are $- 11, 5, 15, - 1, 2$ .

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