- \n
- Similar Triangles<\/a><\/li>\n
- Congruency of Triangles<\/a><\/li>\n
- Basic Proportionality Theorem<\/a><\/li>\n<\/ul>\n
Midpoint Theorem – Definition<\/h2>\n
The term midpoint means the ‘center’ of something, it is the midway between the two endpoints. The midpoint theorem expresses that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side of the triangle. A midpoint generally means a point on a line segment that has an equal distance from both endpoints and bisects the line segment.<\/p>\n
Statement of Midpoint Theorem<\/h3>\n
The midpoint theorem states that \u201cThe line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side of the triangle and is also half of the length of the third side.\u201d<\/p>\n
![\"Midpoint](\"https:\/\/eurekamathanswerkeys.com\/wp-content\/uploads\/2021\/10\/Midpoint-Theorem-Statement-300x285.jpg\")
<\/p>\nProof of Midpoint Theorem<\/h3>\n
Consider the triangle PQR, as shown in the above figure<\/p>\n
Let S and T be the midpoint of the sides PQ and PR. Now, the line ST is said to be parallel to the side QR, where the ST is half of the side QR, i.e.,<\/p>\n
ST \u2225 QR<\/p>\n
ST = (\u00bd \u00d7 QR)<\/p>\n
Now, if you consider the below figure, it is an extended version of the above figure provided.<\/p>\n
![\"Midpoint](\"https:\/\/eurekamathanswerkeys.com\/wp-content\/uploads\/2021\/10\/Midpoint-Theorem-Proof-300x246.jpg\")
<\/p>\nConstruction:<\/h4>\n
Extend the line segment ST and produce it to point O such that, OT = ST.<\/p>\n
In a triangle, PST and TOR,<\/p>\n
TR = PT – – (given)<\/p>\n
\u2220RTO = \u2220PTD (vertically opposite angles)<\/p>\n
\u2220SPT = \u2220TRO (alternate angles)<\/p>\n
By ASA congruence criterion,<\/p>\n
\u25b3TOR \u2245 \u25b3PST<\/p>\n
Therefore,<\/p>\n
\u2220ROT = \u2220PST { by c.p.c.t.c}<\/p>\n
\u2220ORT = \u2220SPT { by c.p.c.t.c}<\/p>\n
and RO = PS { by c.p.c.t.c}<\/p>\n
\u2220ROT and \u2220PSE are the alternate interior angles.<\/p>\n
Assume OR and PQ\u00a0 as two lines that are intersected by the transversal SO.<\/p>\n
Similarly, \u2220ORT and \u2220SPT are the alternate interior angles.<\/p>\n
Let us again assume that OR and PQ are the two lines that are intersected by the transversal PR.<\/p>\n
Therefore, OR \u2225 PQ<\/p>\n
So, OR \u2225\u00a0SQ<\/p>\n
and OR = SQ (since SQ = SP, it is proved that OR = PS)<\/p>\n
Thus, OSQR forms a parallelogram.<\/p>\n
By the properties of a parallelogram, we can write<\/p>\n
QR \u2225 ST<\/p>\n
and QR = ST<\/p>\n
ST \u2225 QR<\/p>\n
and ST = (\u00bd \u00d7 QR)<\/p>\n
Hence, the midpoint theorem is proved.<\/p>\n
Midpoint Theorem Formula<\/h3>\n
In geometry, the midpoint theorem describes the midpoint of the line segment. It specifies the coordinate points of the midpoint of the line segment and can be identified by taking the average of the coordinates of the given endpoints. The formula for midpoint is used to identify the midpoint between the two given points.<\/p>\n
If (x1<\/sub>, y1<\/sub>) and (x2<\/sub>, y2<\/sub>) are the coordinates of two given endpoints, then the midpoint formula is as follows<\/p>\n
Midpoint (x,y) = [ (x1<\/sub>+x2<\/sub>)\/2, (y1<\/sub>+y2<\/sub>)\/2 ]<\/strong><\/em><\/p>\n
Here, x1\u00a0<\/sub>and x2<\/sub> are coordinates of the x-axis, and y1<\/sub> and y2<\/sub> are the coordinates of the y-axis.<\/p>\n
For example, to find the midpoint of (x,y,z) then the midpoint formula is<\/p>\n
(x,y,z) =<\/strong> [ (x1<\/sub>+x2<\/sub>)\/2, (y1<\/sub>+y2<\/sub>)\/2 , (z1<\/sub>+z2<\/sub>)\/2 ]<\/strong><\/em><\/p>\n
The coordinates are (x1<\/sub>,y1<\/sub>,z1<\/sub>), (x2<\/sub>,y2<\/sub>,z2<\/sub>).<\/p>\n
Examples of Midpoint Theorem<\/h3>\n
Example 1:\u00a0<\/strong>
\nIn triangle PQR, the midpoints of\u00a0 PQ, RP, QR are S, T, and U respectively. Find the value of ST, if the value of QR = 12 cm.<\/p>\n![\"Example1](\"https:\/\/eurekamathanswerkeys.com\/wp-content\/uploads\/2021\/10\/Example1-Midpoint-Theorem-300x288.jpg\")
<\/p>\nSolution:\u00a0<\/strong>
\nGiven QR = 14 cm
\nIf S is the midpoint of PQ and T is the midpoint of PR then using the midpoint theorem
\nST = (\u00bd \u00d7 QR)
\nSubstituting the value of QR,
\nST = (\u00bd \u00d7 12)
\nST = 6 cm.
\nTherefore, the value of ST is 6cm<\/strong>.<\/p>\nExample 2:
\n<\/strong>How do you solve a problem such that the point (1,4) is the midpoint of (2,-4), and at what point is the second coordinate?
\nSolution:\u00a0<\/strong>
\nThe first coordinates are (2,-4).
\nLet us assume that the other coordinates are (m,n).
\nNow, the formula for midpoint is
\n(x,y) = [ (x1<\/sub>+x2<\/sub>)\/2, (y1<\/sub>+y2<\/sub>)\/2 ]<\/strong><\/em>
\nSubstitute the above given values,
\nWe get,
\n(1,4) = [(2+m)\/2,(-4+n)\/2]
\n\u21d2 (2+m)\/2 = 1 and (-4+n)\/2 = 4
\n\u21d2 m= 0 and n= 12.
\nTherefore, the second point has coordinates (0,12)<\/strong>.<\/p>\nExample 3:
\n<\/strong>Find the coordinates of the center of the circle. The endpoints of the triangle are given as (6, 2), and (2, 4).
\nSolution:<\/strong>
\nGiven coordinates of the triangle are
\n(6, 2) = (x1<\/sub>, y1<\/sub>) and (2, 4) = (x2<\/sub>, y2<\/sub>)
\nCoordinates of the center of the circle (x,y) = [ (x1<\/sub>+x2<\/sub>)\/2, (y1<\/sub>+y2<\/sub>)\/2 ]
\n<\/strong><\/em>[ (x1<\/sub>+x2<\/sub>)\/2, (y1<\/sub>+y2<\/sub>)\/2 ]
\n=\u00a0<\/strong><\/em>[ ( 6+2)\/2, (2+4)\/2 ]
\n=\u00a0<\/strong><\/em>[ 8\/2, 6\/2 ]
\n=\u00a0<\/strong><\/em>(4, 2)
\nHence, the coordinates of the center of the circle are (4,2)<\/strong>.<\/p>\nFAQs on Midpoint Theorem<\/h3>\n
1. what does the midpoint theorem says?<\/strong>
\nThe midpoint defines that “the line segment drawn from the midpoint of two sides of the triangle is parallel to the third side of the triangle and it is half of the length of the third side of the triangle”.<\/p>\n2. Why is important to define a midpoint?<\/strong>
\nThe midpoint formula applies when one is needed to find the exact center point between two defined points. A line segment has to use this formula to calculate the exact point that bisects a line segment defined by the two given points.<\/p>\n3. How to prove the midpoint theorem?<\/strong>
\nTo prove the midpoint theorem, we use congruency rules. We draw a triangle outside to the given triangle so that it touches the sides of the triangle. Now, we prove that it is congruent to any one part of the triangle. Here, we use the CPCTC rules to prove the equality between the sides of the triangle.<\/p>\n4. What is the use of the midpoint theorem in everyday life?<\/strong>
\nThe midpoint theorem is basically used to know the specific information regarding the length of the sides of the triangle.<\/p>\n","protected":false},"excerpt":{"rendered":"In this article, you can know about the Midpoint Theorem which is invented by Rene Descartes, the famous mathematician. In geometry, the midpoint theorem benefits to find the missing values of the sides of the triangle. The midpoint theorem is one of the important theorems in the concept of geometry. The midpoint theorem deals with … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143388"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=143388"}],"version-history":[{"count":2,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143388\/revisions"}],"predecessor-version":[{"id":143390,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143388\/revisions\/143390"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=143388"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=143388"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=143388"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Congruency of Triangles<\/a><\/li>\n