In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse.
Solution:
Consider a right triangle ABC right angled at B.
D is the midpoint of the hypotenuse AC
Extend BD upto E so that BD = DE and join CE
We have to prove that the line segment joining the midpoint of the hypotenuse to the opposite vertex is half the hypotenuse i.e.,BD = 1/2 AC
Considering triangles ADB and CDE,
Given, D is the midpoint of AC
So, AC = CD
Also, BD = DE
We know that the vertically opposite angles are equal.
∠ADB = ∠CDE
SAS criterion states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are congruent.
By SAS criterion, the triangles ADB and CDE are congruent.
By CPCTC,
AB = EC
∠BAD = ∠DCE
Where ∠BAD and ∠DCE are alternate angles.
This implies that EC is parallel to AB and BC is a transversal.
We know that if two lines are parallel and cut by a transversal, the sum of interior angles lying on the same side of the transversal is always supplementary.
∠ABC + ∠BCE = 180°
90 + ∠BCE = 180°
∠BCE = 180° - 90°
∠BCE = 90°
Considering triangles ABC and ECB,
We know AB = EC
Common side = BC
∠ABC = ∠ECB = 90°
By SAS criterion, the triangles ABC and ECB are congruent
By CPCT,
AC = EB
Dividing by 2 on both sides,
AC/2 = EB/2
We know BD = DE
So, BE = BD + DE
BE = BD + BD
BE = 2BD
BD = BE/2
Now, AC/2 = BD
Therefore, BD = 1/2 AC
✦ Try This: In right angled triangle ABC, right angled at C,M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM=CM. Point D is joined to point B. Show that: ∠DBC is a right angle.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.4 Problem 14
In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse
Summary:
In a right triangle, it is proven that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse by CPCT
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