QuestionFind the coordinates of point $C$ in right isosceles triangle $\triangle ABC$ where $A(2,3)$ and $B(7,3)$ .

Studdy Solution

STEP 1

Assumptions1. $\overline{\mathrm{AB}}$ is the hypotenuse of the right isosceles $\triangle \mathrm{ABC}$ . . The coordinates of point A are (,3).
3. The coordinates of point B are (7,3).
4. In a right isosceles triangle, the two legs are of equal length and the angles are45°,45°, and90°.

STEP 2

First, we need to find the length of the hypotenuse $\overline{\mathrm{AB}}$ . We can do this using the distance formula.
$AB = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$

STEP 3

Now, plug in the given values for the coordinates of A and B to calculate the length of AB.
$AB = \sqrt{(7 -2)^2 + (3 -3)^2}$

STEP 4

Calculate the length of AB.
$AB = \sqrt{()^2 + (0)^2} = \sqrt{25} =$

STEP 5

In a right isosceles triangle, the length of the hypotenuse is $\sqrt{2}$ times the length of each leg. Therefore, the length of AC and BC (the legs of the triangle) is the length of the hypotenuse divided by $\sqrt{2}$ .
$AC = BC = \frac{AB}{\sqrt{2}}$

STEP 6

Plug in the value for the length of AB to calculate the length of AC and BC.
$AC = BC = \frac{5}{\sqrt{2}}$

STEP 7

implify the length of AC and BC.
$AC = BC = \frac{5\sqrt{2}}{2}$

STEP 8

Since $\triangle ABC$ is a right isosceles triangle and AB is the hypotenuse, the coordinates of C will be either above or below the line AB. We can find these coordinates by adding or subtracting the length of AC (or BC) from the y-coordinate of A (or B).
$C1 = (x1, y1 \pm AC)$ $C2 = (x2, y2 \pm BC)$

STEP 9

Plug in the values for the coordinates of A and B and the length of AC and BC to calculate the coordinates of C.
$C = (2,3 \pm \frac{5\sqrt{2}}{2})$ $C2 = (7,3 \pm \frac{5\sqrt{2}}{2})$

STEP 10

Calculate the coordinates of C.
$C = (2,3 + \frac{5\sqrt{2}}{2}), (2,3 - \frac{5\sqrt{2}}{2})$ $C2 = (7,3 + \frac{5\sqrt{2}}{2}), (7,3 - \frac{5\sqrt{2}}{2})$ However, since $\overline{\mathrm{AB}}$ is the hypotenuse, we discard the points $(2,3 + \frac{5\sqrt{2}}{2})$ and $(7,3 + \frac{5\sqrt{2}}{2})$ .
The possible coordinates of C are $(2,3 - \frac{5\sqrt{2}}{2})$ and $(7,3 - \frac{5\sqrt{2}}{2})$ .

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