Problem 38 Find the lengths of the medians ... [FREE SOLUTION]

Chapter 1: Problem 38

Find the lengths of the medians of the triangle with vertices \(A(1,0), B(3,6),\) and \(C(8,2) .\) (A median is a line segment from a vertex to the midpoint of the opposite side.)

Short Answer

Expert verified

AM = \(\frac{\sqrt{145}}{2}\), BM = \(\sqrt{27.25}\), CM = \(\sqrt{37}\).

Step by step solution

Find Midpoint of BC

To find the midpoint of segment BC, use the midpoint formula: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Plugging in the coordinates for B(3,6) and C(8,2):\[ \text{Midpoint of } BC = \left( \frac{3 + 8}{2}, \frac{6 + 2}{2} \right) = \left( \frac{11}{2}, 4 \right) \]

Calculate Median from A to BC

Use the distance formula to calculate the length of median AM (from A to midpoint M of BC):\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Applying it to A(1,0) and M\left( \frac{11}{2}, 4 \right):\[ AM = \sqrt{\left( \frac{11}{2} - 1 \right)^2 + (4 - 0)^2} \]\[ = \sqrt{\left( \frac{9}{2} \right)^2 + 4^2} \]\[ = \sqrt{\frac{81}{4} + 16} \]\[ = \sqrt{\frac{145}{4}} \]\[ = \frac{\sqrt{145}}{2} \]

Find Midpoint of AC

To find the midpoint of segment AC, use the midpoint formula:\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Plugging in coordinates for A(1,0) and C(8,2):\[ \text{Midpoint of } AC = \left( \frac{1 + 8}{2}, \frac{0 + 2}{2} \right) = (4.5, 1) \]

Calculate Median from B to AC

Use the distance formula to calculate the length of median BM (from B to midpoint M of AC):\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Applying it to B(3,6) and M(4.5,1):\[ BM = \sqrt{(4.5 - 3)^2 + (1 - 6)^2} \]\[ = \sqrt{1.5^2 + (-5)^2} \]\[ = \sqrt{2.25 + 25} \]\[ = \sqrt{27.25} \]

Find Midpoint of AB

To find the midpoint of segment AB, use the midpoint formula:\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Plugging in coordinates for A(1,0) and B(3,6):\[ \text{Midpoint of } AB = \left( \frac{1 + 3}{2}, \frac{0 + 6}{2} \right) = (2,3) \]

Calculate Median from C to AB

Use the distance formula to calculate the length of median CM (from C to midpoint M of AB):\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Applying it to C(8,2) and M(2,3):\[ CM = \sqrt{(2 - 8)^2 + (3 - 2)^2} \]\[ = \sqrt{(-6)^2 + 1^2} \]\[ = \sqrt{36 + 1} \]\[ = \sqrt{37} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Midpoint Formula

The midpoint formula is a crucial tool in geometry for finding the center point between two points. When you have two coordinates, let's say

Point 1: \(x_1, y_1\)
Point 2: \(x_2, y_2\)

You can find the midpoint by using the formula:

Midpoint = \( rac{x_1 + x_2}{2}, rac{y_1 + y_2}{2} \)

This formula works by averaging the x-coordinates and the y-coordinates separately to find their central point. This concept is extremely useful in problems involving line segments, such as finding medians in triangles. Medians in a triangle are the line segments that connect a vertex to the midpoint of the opposite side. Knowing how to find midpoints makes it much easier to correctly identify these medians.

Distance Formula

The distance formula helps us determine the length of a line segment between two points in a coordinate plane. It is derived from the Pythagorean theorem. Given two points,

Point 1: \(x_1, y_1\)
Point 2: \(x_2, y_2\)

To find the distance between them, you use:

Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\

This formula calculates the hypotenuse of a right triangle formed by the differences in x-values and y-values as its legs. In geometry, especially involving triangles, the distance formula is invaluable. You can use it not only to compute line lengths directly related to medians but also in broader applications where measuring precise distances is necessary, enhancing your understanding of geometric constructs.

Geometry

Geometry provides the tools to understand shapes, sizes, and the properties of space. In the context of triangles, a key aspect includes understanding medians 鈥� line segments that connect a vertex with the midpoint of the opposite side. These medians are crucial because:

They divide a triangle into two smaller triangles of equal area.
The point where all three medians intersect is known as the centroid, which is the triangle鈥檚 center of balance.

With an understanding of medians, one can solve various problems; from locating centroids to determining area-related properties. Medians provide a deeper insight into the geometric balance and symmetry inherent in triangles, adding an extra layer to your spatial reasoning capabilities.

91影视

Find the lengths of the medians of the triangle with vertices \(A(1,0), B(3,6),\) and \(C(8,2) .\) (A median is a line segment from a vertex to the midpoint of the opposite side.)

Short Answer

Step by step solution

Find Midpoint of BC

Calculate Median from A to BC

Find Midpoint of AC

Calculate Median from B to AC

Find Midpoint of AB

Calculate Median from C to AB

Key Concepts

Midpoint Formula

Distance Formula

Geometry

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