Maharashtra Board Class 8 Maths part 1 Chapter 4 Altitudes and Medians of a triangle PDF Download

Altitudes and Medians of a Triangle

Let's Recall

In the previous standard we have learnt that the bisectors of angles of a triangle, as well as the perpendicular bisectors of its sides are concurrent. These points of concurrence are respectively called the incentre and the circumcentre of the triangle.

Activity

Draw a line. Take a point outside the line. Draw a perpendicular from the point to the line with the help of a set - square.

Altitude

The perpendicular segment drawn from a vertex of a triangle on the side opposite to it is called an altitude of the triangle. In triangle ABC, seg AP is an altitude on the base BC.

To Draw Altitudes of a Triangle

1. Draw any triangle XYZ.

2. Draw a perpendicular from vertex X on the side YZ using a set - square. Name the point where it meets side YZ as R. Seg XR is an altitude on side YZ.

3. Considering side XZ as a base, draw an altitude YQ on side XZ. seg YQ is perpendicular to seg XZ.

4. Consider side XY as a base, draw an altitude ZP on seg XY. seg ZP is perpendicular to seg XY.

Seg XR, seg YQ, seg ZP are the altitudes of triangle XYZ.

Note that, the three altitudes are concurrent. The point of concurrence is called the orthocentre of the triangle. It is denoted by the letter 'O'.

Teacher's Note

An altitude is a line from a corner straight down to the opposite side at a right angle. Like when you drop a weight from the top of a building straight down, it is like an altitude of a triangle.

Exam Trick

Remember: Altitude = height. It is always perpendicular (90 degrees) to the base. Just like how your height is measured straight up from the ground.

Points to Remember

An altitude goes from a vertex to the opposite side. It always makes a 90 degree angle. All three altitudes meet at one point called the orthocentre. This point 'O' can be inside, outside, or on the triangle depending on the type of triangle.

The Location of the Orthocentre of a Triangle

Activity I

Draw a right angled triangle and draw all its altitudes. Write the point of concurrence.

Activity II

Draw an obtuse angled triangle and all its altitudes. Do they intersect each other? Draw the lines containing the altitudes. Observe that these lines are concurrent.

Activity III

Draw an acute angled triangle ABC and all its altitudes. Observe the location of the orthocentre.

The altitudes of a triangle pass through exactly one point; that means they are concurrent. The point of concurrence is called the orthocentre and it is denoted by 'O'.

The orthocentre of a right angled triangle is the vertex of the right angle.

The orthocentre of an obtuse angled triangle is in the exterior of the triangle.

The orthocentre of an acute angled triangle is in the interior of the triangle.

Teacher's Note

The location of the orthocentre changes based on the type of triangle. In a right angle triangle, it is at the corner where the right angle is. This is important to remember for your exam.

Exam Trick

Remember: Right angle = orthocentre at the right angle vertex. Acute triangle = orthocentre inside. Obtuse triangle = orthocentre outside. Make a quick diagram to remember.

Points to Remember

All three altitudes of a triangle meet at the orthocentre. In a right triangle, the orthocentre is at the right angle corner. In an acute triangle, the orthocentre is inside the triangle. In an obtuse triangle, the orthocentre is outside the triangle.

Median

The segment joining the vertex and midpoint of the opposite side is called a median of the triangle. In triangle HCF, seg FD is a median on the base CH.

To Draw Medians of a Triangle

1. Draw triangle ABC.

2. Find the mid-point P of side AB. Draw seg CP.

3. Find the mid-point Q of side BC. Draw seg AQ.

4. Find the mid-point R of side AC. Draw seg BR.

Seg PC, seg QA and seg BR are medians of triangle ABC. Note that the medians are concurrent. Their point of concurrence is called the centroid. It is denoted by G.

Activity IV

Draw three different triangles; a right angled triangle, an obtuse angled triangle and an acute angled triangle. Draw the medians of the triangles. Note that the centroid of each of them is in the interior of the triangle.

The Property of the Centroid of a Triangle

Draw a sufficiently large triangle ABC.

Draw medians; seg AR, seg BQ and seg CP of triangle ABC.

Name the point of concurrence as G.

Measure the lengths of segments from the figure and fill in the boxes in the following table.

Observe that all of these ratios are nearly 2:1.

The medians of a triangle are concurrent. Their point of concurrence is called the Centroid and it is denoted by G. For all types of triangles the location of G is in the interior of the triangles. The centroid divides each median in the ratio 2:1.

Teacher's Note

A median is a line from a corner to the middle point of the opposite side. Like when you find the centre point of a seesaw, you are finding something like a median. The centroid is where all medians meet, and it always stays inside the triangle.

Exam Trick

Remember: Median = middle point. The centroid divides each median in the ratio 2:1 from the vertex. This means the part from vertex to centroid is twice as long as the part from centroid to the midpoint.

Points to Remember

A median connects a vertex to the middle point of the opposite side. All three medians meet at a point called the centroid, shown as 'G'. The centroid is always inside any triangle. The centroid divides each median into two parts with a ratio of 2:1 from the vertex.

Let's Discuss

As shown in the adjacent figure, a student drew triangle ABC using five parallel lines of a note book. Then he found the centroid G of the triangle. How will you decide whether the location of G he found, is correct.

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