Given points A(1,1), B(13,9), and C(3,7) . D is the midpoint of A B, and E is the midpoint of

step 1 Okay, 1 plus 13 over 2, comma 1 plus 9 over 2 gives us 7 .5. For part B, the slope, Y 2 minus Y

Given points A(1,1), B(13,9), and C(3,7) . D is the...

Given points $A(1,1), B(13,9),$ and $C(3,7) . \quad D$ is the midpoint of $\overline{A B},$ and $E$ is the midpoint of $\overline{A C}$ . a. Find the coordinates of $D$ and $E$ b. Use slopes to show that $\overline{D E} \| \overline{B C}.$ c. Use the distance formula to show that $D E=\frac{1}{2} B C .$

Given points $A(1,1), B(13,9),$ and $C(3,7) . \quad D$ is the midpoint of $\overline{A B},$ and $E$ is the midpoint of $\overline{A C}$ .
a. Find the coordinates of $D$ and $E$
b. Use slopes to show that $\overline{D E} \| \overline{B C}.$
c. Use the distance formula to show that $D E=\frac{1}{2} B C .$

Key Concepts

Triangle Midsegment Theorem

The Triangle Midsegment Theorem states that the segment joining the midpoints of two sides of a triangle is both parallel to the third side and half its length. This theorem is key in many proofs and problems in geometry because it allows one to deduce relationships of parallelism and proportionality without requiring additional constructions.

Distance Formula

The distance formula is used to calculate the length of a line segment between two points in the coordinate plane. It is derived from the Pythagorean theorem and given two points (x1, y1) and (x2, y2), the distance is determined by sqrt((x2 - x1)^2 + (y2 - y1)^2). This formula is particularly useful in verifying relationships between segments, such as showing one segment is a fraction of another.

Midpoint Formula

The midpoint formula is used to find the point that is exactly halfway between two given points in the coordinate plane. This concept is essential in many geometry problems because it allows one to divide segments into two equal parts. In coordinate geometry, if you have two points with coordinates (x1, y1) and (x2, y2), their midpoint is calculated as ((x1 + x2)/2, (y1 + y2)/2).

Slope of a Line

The concept of the slope of a line determines how steep the line is and in which direction it tilts. In coordinate geometry, the slope is computed as the ratio of the difference in the y-coordinates to the difference in the x-coordinates of two points on the line. This is fundamental in establishing parallelism because lines that are parallel have equal slopes.

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