An isosceles triangle is a triangle that contains two sides of equal length, two angles opposite to those sides are equal in measure. We have provided proof for the many isosceles theorems and statements. Similarly, the proof for the statement straight lines joining the extremities of the base of an isosceles triangle to the midpoint of the opposite side are equal is given here.
Straight Lines Joining the Extremities of the Base of an Isosceles Triangle
Here we will prove that the straight lines joining the extremities of the base of an isosceles triangle to the midpoint of the opposite side are equal.
Let us take ∆XYZ, where XY = XZ
M, N are the midpoints of XY and XZ respectively
To prove: ZN = YM
Proof:
| Statement | Reason |
|---|---|
| XY = XZ | Given |
| NY = ½XY | N is the midpoint of XY |
| MZ = ½XZ | M is the midpoint of XZ |
| NY = MZ | From statements 1, 2, and 3 |
| In ∆NYZ and ∆MYZ, NY = MZ YZ = YZ ∠XYZ = ∠XZY |
From statement 4 Common side XY = XZ |
| ∆NYZ ≅ ∆MYZ | By SAS Criterion |
| ZN = YM | CPCTC |
Hence proved.
Also, Check
- If Two Isosceles Triangles on Same Base, Legs of One Triangle are Twice of Legs of Other
- Bisectors of the Angles of a Triangle Meet at a Point
Example Questions with Solutions
Question 1:
If △ABC is an isosceles triangle with base BC. The length of the median drawn from vertex C to the opposite side is 4 cm, then find what is the length of the median drawn from another vertex of the base.
Solution:
Given that,
△ABC is an isosceles triangle with base BC
The length of the median drawn from C to its opposite side = 4 cm
We know that straight lines joining the extremities of the base of an isosceles triangle to the midpoint of the opposite side are equal.
So, the length of another median drawn from vertex B to the opposite side = 4 cm
Question 2:
In an isosceles triangle, the length of the straight line drawn from one extreme base to the midpoint of the opposite side is 15 ft. Find the length of the line drawn from another base extreme.
Solution:
The length of the straight line extreme base to the midpoint of the opposite side = 15 ft
According to the isosceles triangle theorem,
The length of the straight line drawn from one extreme base to the midpoint of the opposite side = The length of a straight line drawn from the second extreme base to the midpoint of its opposite side
Therefore, the length of the line is drawn from another base extreme to the opposite side = 15 ft.
Frequently Asked Question’s
1. What is the base of an isosceles triangle?
In an isosceles triangle, the two equal sides are called legs and the third side is called the base.
2. What is the rule for an isosceles triangle?
The rule for an isosceles triangle is that the triangle must have two sides of equal length. The isosceles triangle theorem states that the angles opposite to the equal sides are also equal.
3. What is the median of a triangle?
Median of a triangle is the straight line that joins vertex to the midpoint of the side which is opposite to the respective vertex.