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.
What is 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.
Application of the Midpoint Theorem
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?
Over 79,000 lessons in all major subjects
Get access risk-free for 30 days,
just create an account.
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:
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.
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
Did you know… We have over 200 college
courses that prepare you to earn
credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the
first two years of college and save thousands off your degree. Anyone can earn
credit-by-exam regardless of age or education level.