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High School Geometry: Help and Review13 chapters | 162 lessons

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Lesson Transcript

Instructor:
*Ellen Manchester*

Postulates and proven theorems allow us to know and understand certain things about geometric figures. In this lesson, we will be learning about the Midpoint Theorem.

A **midpoint** is a point on a line segment equally distant from the two endpoints. The Midpoint Theorem is used to make a bold statement regarding triangle sides and their lengths. Given a triangle, if we connect two sides with a line segment, and this line segment joins each of the two sides at the centers, or midpoints of each side, we can know two very important aspects about the triangle and the relationships between the sides.

**The Midpoint Theorem** states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

Anytime you have a line segment that connects two sides of a triangle at the midpoints, you automatically know that the sides are cut in half, and that the segment is parallel to the third side of the triangle. Parallel sides are shown by using this symbol ||. You also know the line segment is one-half the length of the third side.

Take a look at this figure:

This indicates that points *R* and *S* are midpoints of sides *AT* and *AV*, respectively. From the Midpoint Theorem, since the segment *RS* connects the two sides at the midpoints, then *RS* || *TV* and *RS* is one-half the length of side *TV*.

This theorem allows us to prove some things about the triangle. First, if we know the length of *TV*, then we can figure out the length of *RS*, and vice-versa, since *RS* = ½(*TV*). It also allows us to find the lengths of *AS, VS, TR* and *AR*. Since *RS* is parallel to *TV*, then we also know the distance between these two line segments are equal.

Let's see what we can find out about triangle *BCD* and each of its sides. Given that line segment *TS* connects the two sides *BC* and *DC* at the respective midpoints, *T* and *S*.

- If
*BD*= 26*cm*, find the length of segment*TS*. - If
*BT*= 15*cm*, what is the length of segment*CT*? - If
*CD*= 20*cm*, what are the lengths of both segments*SD*and*CS*?

- If
*BD*= 26*cm*,*TS*is half this length.*TS*= 1/2 x 26 = 13*cm* - If
*BT*= 15*cm*, and given*T*is the midpoint of*BC*, then*CT*is also 15*cm*. - If
*CD*= 20*cm*, and given*S*is the midpoint of*CD*, then*CS*and*SD*are both 10*cm*.

Let's try another:

Given:

*YZ || WS**Y*and*Z*are midpoints of*WX*and*SX*, respectively.*WY*= 7*cm**XS*= 18*cm**YZ*= 10*cm*

What can you tell about *XY*, *ZS* and *WS*?

Since *Y* is the midpoint of *XW* and *WY* = 7 *cm*, then *XY* must also be 7 *cm*.

Since *XS* = 18 *cm* and *Z* is the midpoint of *XS*, then *ZS* must be half of *XS*, so *ZS* = 9 *cm*.

Using the Midpoint Theorem, since *Y* and *Z* are midpoints and *YZ* || *WS*, we know that *YZ* is one-half the length of *WS*. *YZ* is 10 *cm*, so twice that amount is 20 *cm*. *WS* = 20 *cm*.

Let's review:

**The Midpoint Theorem** is used to find specific information regarding lengths of sides of triangles. The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

We can use this information to find the lengths of the sides of the triangle.

**Midpoint** - a point on a line segment equally distant from the two endpoints

**Midpoint Theorem** - theorem which states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side

Determine whether you can do the following after reviewing this lesson on the Midpoint Theorem:

- Locate the midpoint on a line segment
- State the Midpoint Theorem
- Find the length of the side of a triangle using the Midpoint Theorem

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High School Geometry: Help and Review13 chapters | 162 lessons

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