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The Triangles and its Properties

    Two Special Triangles : Equilateral and Isosceles

    Exercise 6.1

    What Have We Discussed ?

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7th class > The Triangles and its Properties > Exercise 6.1

Exercise 6.1

1. In ∆ PQR, D is the mid-point of QR.

∆ PQR

PM is ?

PD is ?

Is QM = MR ?

Solution:

PM is on QR.

Also, PD divides QR into equal parts as D is the mid-point of .

Therefore, PM is .

Also, D is the mid-point of .

Therefore, QD = DR and hence PD is the

Is QM = MR ? ! QM ≠ MR because D is the mid-point of QR.

2. Draw rough sketches for the following:

a

(a) In ∆ABC, BE is a median.

Solution:

A median of a triangle is a line segment that is drawn from a vertex to the opposite side of the vertex, and it divides the opposite side into two parts. The rough sketch of ▲ABC, where BE is a , is drawn below.

∆ABC

b

(b) In ∆PQR, PQ and PR are altitudes of the triangle.

Solution:

An altitude of a triangle is defined as a drawn from the vertex to the line containing the opposite side of the triangle. The rough sketch of ∆PQR, where and are the altitudes of the triangle is drawn below.

∆PQR

c

(c) In ∆XYZ, YL is an altitude in the exterior of the triangle.

Solution:

An altitude of a triangle is defined as a perpendicular drawn from the vertex to the line containing the opposite side of the triangle. The rough sketch of ∆XYZ, YL is an altitude in the of the triangle is drawn below.

∆XYZ

3. Verify by drawing a diagram if the median and altitude of an isosceles triangle can be same.

∆XYZ

Solution:

Given that the triangle is an triangle.

It is required to verify if the median and altitude of the given triangle can be the same. To do this, construct an isosceles triangle. An isosceles triangle has two sides.

Construct an isosceles triangle △ABC with sides AB = AC and draw a median AL that divides the base of the triangle into equal parts.

From the triangle, it can be seen that the median makes a ° angle with the base .

So, AL is the of the triangle △ABC.

Hence verified, AL is the median and altitude of the given triangle △ABC.