Problem 26 - Precalculus

Question

The vertices of \(\triangle A B C\) are \(A(1,1), B(9,3),\) and \(C(3,5)\) (a) Find the perimeter of \(\triangle A B C\) (b) Find the perimeter of the triangle that is formed by joining the midpoints of the three sides of \(\triangle A B C\) (c) Compute the ratio of the perimeter in part (a) to the perimeter in part (b). (d) What theorem from geometry provides the answer for part (c) without using the results in (a) and (b)?

Step-by-Step Solution

Answer

(a) \(\sqrt{68} + \sqrt{40} + \sqrt{20}\); (b) \(\sqrt{5} + \sqrt{17} + \sqrt{10}\); (c) 2; (d) Midsegment Theorem.

Step 1: Calculate Side Lengths of Triangle ABC

To find the perimeter of \(\triangle ABC\), we first need to calculate the lengths of the sides \(AB\), \(BC\), and \(CA\) using the distance formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).\(AB = \sqrt{(9-1)^2 + (3-1)^2} = \sqrt{64 + 4} = \sqrt{68}\).\(BC = \sqrt{(9-3)^2 + (3-5)^2} = \sqrt{36 + 4} = \sqrt{40}\).\(CA = \sqrt{(3-1)^2 + (5-1)^2} = \sqrt{4 + 16} = \sqrt{20}\).

Step 2: Calculate Perimeter of Triangle ABC

Now, add the lengths of sides \(AB\), \(BC\), and \(CA\) to find the perimeter: \(\text{Perimeter of } \triangle ABC = \sqrt{68} + \sqrt{40} + \sqrt{20}\).

Step 3: Find Midpoints of Triangle ABC's Sides

Find the midpoints of \(AB\), \(BC\), and \(CA\):Midpoint of \(AB\): \((\frac{1+9}{2}, \frac{1+3}{2}) = (5, 2)\).Midpoint of \(BC\): \((\frac{9+3}{2}, \frac{3+5}{2}) = (6, 4)\).Midpoint of \(CA\): \((\frac{3+1}{2}, \frac{5+1}{2}) = (2, 3)\).

Step 4: Calculate Side Lengths of Midpoint Triangle

Find the lengths of the sides of the triangle formed by midpoints:Using the distance formula, calculate:\(MP_{AB-BC} = \sqrt{(6-5)^2 + (4-2)^2} = \sqrt{1 + 4} = \sqrt{5}\).\(MP_{BC-CA} = \sqrt{(6-2)^2 + (4-3)^2} = \sqrt{16 + 1} = \sqrt{17}\).\(MP_{CA-AB} = \sqrt{(2-5)^2 + (3-2)^2} = \sqrt{9 + 1} = \sqrt{10}\).

Step 5: Calculate Perimeter of Midpoint Triangle

Add up the lengths of the sides of the midpoint triangle to find its perimeter:\(\text{Perimeter of Midpoint Triangle} = \sqrt{5} + \sqrt{17} + \sqrt{10}\).

Step 6: Compute Ratio of Perimeters

Calculate the ratio of the perimeter of \(\triangle ABC\) to the perimeter of the midpoint triangle:\( \text{Ratio} = \frac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of Midpoint Triangle}} = \frac{\sqrt{68} + \sqrt{40} + \sqrt{20}}{\sqrt{5} + \sqrt{17} + \sqrt{10}}\approx 2\).

Step 7: Identify Theorem

The theorem that states the ratio of the perimeter of a triangle to the perimeter of the triangle formed by joining its midpoints is 2:1 is the Midsegment Theorem or the Triangle Midsegment Theorem. This theorem implies that any segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long, leading to a 2:1 perimeter ratio.

Key Concepts

Distance Formula

The Distance Formula is essential for calculating the distance between two points on a coordinate plane. This formula is derived from the Pythagorean Theorem. It is often used in geometry to find the lengths of the sides of polygons, especially triangles.
To use the Distance Formula, you need the coordinates of the two points:

• Let's say you have points \(A(x_1, y_1)\) and \(B(x_2, y_2)\).
• The distance \(d\) between them is given by: \(\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]\)

Applying this formula helps determine side lengths such as \(AB\), \(BC\), and \(CA\) in \(\triangle ABC\).
For example, the calculation for side \(AB\) of \(\triangle ABC\) with vertices \(A(1,1), B(9,3),\) and \(C(3,5)\) is \( \sqrt{68} \).

Midsegment Theorem

The Midsegment Theorem is an intriguing concept in triangle geometry. It states that a line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. This property has practical applications, such as estimating triangle perimeters efficiently and understanding geometric properties better.
For \(\triangle ABC\), the midsegment triangle is formed by connecting these midpoints:

• Midpoint of \(AB\): \( (5, 2) \)
• Midpoint of \(BC\): \( (6, 4) \)
• Midpoint of \(CA\): \( (2, 3) \)

These points create a smaller triangle within the original one, with a distinct relationship as described by the theorem. The length relationships help us verify the 2:1 perimeter ratio without complex calculations.
Despite appearing simple, this theorem deepens our understanding of triangle properties and their symmetrical aspects.

Ratio of Perimeters

Understanding the Ratio of Perimeters helps in comparing different shapes within geometric figures. If you have a triangle and its midsegment triangle, their perimeters are related by a consistent ratio.
In this exercise, the perimeter of \(\triangle ABC\) compared to the perimeter of the midsegment triangle is approximately 2:1. This is no coincidence but a direct implication of the Midsegment Theorem. Calculating the actual perimeters involves:

• The Perimeter of \(\triangle ABC\) as \(\sqrt{68} + \sqrt{40} + \sqrt{20}\)
• The Perimeter of the midsegment triangle as \(\sqrt{5} + \sqrt{17} + \sqrt{10}\)

This 2:1 ratio holds because the midsegment triangle's sides are half the length of the original triangle's sides due to the theorem.
This aspect not only makes calculations easier but also highlights inherent geometric relations in triangle shapes.