In triangle PQR, let X be the intersection of the angle bisector of angle P with side QR, and let Y be the foot of the perpendicular from X to line PR.  If PQ = 9, QR = 9, and PR = 9, then compute the length of XY.

 Jun 5, 2023


 Jun 5, 2023

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 Jun 5, 2023

The angle bisector theorem states that in a triangle, the angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.

In triangle PQR, we have PQ = 9, QR = 9, and PR = 9. Since PR is the longest side, angle P is the largest angle in the triangle.

Let's label the length of PX as a and QX as b. Since the angle bisector of angle P intersects QR at point X, we can use the angle bisector theorem to set up the following equation:

PX / QX = PR / QR

a / b = 9 / 9

a / b = 1

Since a and b are in the ratio of 1:1, we can conclude that PX = QX.

Now, let's consider triangle PXY. Since PX = QX, triangle PXY is an isosceles triangle with PX = QX. Additionally, Y is the foot of the perpendicular from X to line PR, which means XY is the altitude of triangle PXY.

In an isosceles triangle, the altitude from the vertex angle bisects the base. Therefore, XY will bisect PR, and we can conclude that PY = YR.

Since PR = 9, PY + YR = PR implies PY + PY = 9, which simplifies to 2PY = 9.

Thus, PY = YR = 9 / 2 = 4.5.

Finally, we can use the Pythagorean theorem in right triangle PXY to find the length of XY:

XY^2 = PX^2 + PY^2

Since PX = PY = 4.5, we have:

XY^2 = 4.5^2 + 4.5^2

XY^2 = 20.25 + 20.25

XY^2 = 40.5

Taking the square root of both sides, we get:

XY = √40.5

Therefore, the length of XY is approximately 6.36 (rounded to two decimal places).

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 Jun 15, 2023

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