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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.

 Apr 29, 2023
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In the triangle PQR, let X represent the point where angle P and side QR connect, and let Y represent the foot of the perpendicular that runs from X to side PR. The length of XY is  2.25 × √(3) units.

What are triangles?

By drawing straight lines from three non-collinear points, a triangle is a three-sided polygon.

It is a basic geometric shape having several properties and applications in science, technology, engineering, and other fields.

Depending on the size of their angles and side lengths, triangles can be divided into different categories.

Due to the fact that PQ = QR = PR, triangle PQR is an equilateral triangle.

Let's write x as the length of each angle in this triangle. Since the angle P is divided into two equal halves by the angle bisector, the measures of the angles PXQ and QXR are both x/2.

Let's write d to represent XY's length. Trigonometry can be used to determine the length of XY since triangle PXY is a right triangle. Specifically, we have

tan(30) = XY / PY

Since PY = PQ - QY and PQ = QR = 9, we have PY = 9 - QY. Therefore:

tan(30) = XY / (9 - QY)

Solving for XY, we get:

XY = (9 - QY) tan(30)

QY needs to be located. The Pythagorean theorem can be used to determine QY because triangle QYX is also a right triangle:

QY² + XY² = QX²

Triangle QRX being a 30-60-90 triangle gives us:

QX = QR / 2 = 4.5

Therefore:

QY² + XY² = 4.5²

QY² + (9 - QY)² tan²(30) = 4.5²

The result of simplifying and solving for QY is:

QY = 2.25

Inputting this value into the XY expression yields the following result:

XY = (9 - 2.25) tan(30) = 2.25 × √(3)

 May 1, 2023

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